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\\begin{pmatrix} 2-4+6 \\\\ -3+8-2 \\\\ 5+1-4 \\end{pmatrix} = \\begin{pmatrix} 4 \\\\ 3 \\\\ 2 \\end{pmatrix}$ M1\n- Resultant $= 4\\mathbf{i} + 3\\mathbf{j} + 2\\mathbf{k}$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"States or implies the resultant force is the sum of the three individual vectors.","code":"M"},{"id":"p2","marks":1,"description":"Correct calculation showing the addition of components, either as a column vector or sum of i, j, k terms.","code":"M"},{"id":"p3","marks":1,"description":"Correct final resultant force in vector notation.","code":"A"}],"finalAnswers":["4\\mathbf{i} + 3\\mathbf{j} + 2\\mathbf{k}"],"generatedAt":"2026-05-06T14:03:58.209Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"unit_vector","label":"Unit Vector Addition","steps":[{"type":"text","order":0,"title":"Understand the goal","content":"To find the resultant force acting on the boat, we need to sum the three individual force vectors. The formula is:\n\n$$\\mathbf{R} = \\mathbf{F}_1 + \\mathbf{F}_2 + \\mathbf{F}_3$$\n\nIdentifying that the resultant force is found by summing the individual forces earns the first method mark **(M1)**.","highlights":[{"text":"Find the resultant force.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\mathbf{F}_1 = \\htmlData{anchor=f1-i}{\\color{@sky}{2}}\\mathbf{i} \\htmlData{anchor=f1-j}{\\color{@amber}{- 3}}\\mathbf{j} \\htmlData{anchor=f1-k}{\\color{@purple}{+ 5}}\\mathbf{k}$"},{"latex":"$$\\mathbf{F}_2 = \\htmlData{anchor=f2-i}{\\color{@sky}{-4}}\\mathbf{i} \\htmlData{anchor=f2-j}{\\color{@amber}{+ 8}}\\mathbf{j} \\htmlData{anchor=f2-k}{\\color{@purple}{+ 1}}\\mathbf{k}$"},{"latex":"$$\\mathbf{F}_3 = \\htmlData{anchor=f3-i}{\\color{@sky}{+6}}\\mathbf{i} \\htmlData{anchor=f3-j}{\\color{@amber}{- 2}}\\mathbf{j} \\htmlData{anchor=f3-k}{\\color{@purple}{- 4}}\\mathbf{k}$"},{"latex":"$$\\mathbf{R} = (\\htmlData{anchor=sum-i}{\\color{@sky}{2 - 4 + 6}})\\mathbf{i} + (\\htmlData{anchor=sum-j}{\\color{@amber}{-3 + 8 - 2}})\\mathbf{j} + (\\htmlData{anchor=sum-k}{\\color{@purple}{5 + 1 - 4}})\\mathbf{k}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"f1-i"},{"color":"sky","style":"text-color","anchor":"f2-i"},{"color":"sky","style":"text-color","anchor":"f3-i"},{"color":"amber","style":"text-color","anchor":"f1-j"},{"color":"amber","style":"text-color","anchor":"f2-j"},{"color":"amber","style":"text-color","anchor":"f3-j"},{"color":"purple","style":"text-color","anchor":"f1-k"},{"color":"purple","style":"text-color","anchor":"f2-k"},{"color":"purple","style":"text-color","anchor":"f3-k"}],"connections":[{"to":"sum-i","from":"f1-i","color":"sky","label":null,"style":"dashed"},{"to":"sum-j","from":"f1-j","color":"amber","label":null,"style":"dashed"},{"to":"sum-k","from":"f1-k","color":"purple","label":null,"style":"dashed"}]},"order":1,"title":"Group the components","description":"We can directly sum the corresponding coefficients of the $\\mathbf{i}$, $\\mathbf{j}$, and $\\mathbf{k}$ components."},{"type":"text","order":2,"title":"Calculate final vector","content":"Now we simplify the calculation for each component:\n\n- **$\\mathbf{i}$-component:** $2 - 4 + 6 = 4$\n- **$\\mathbf{j}$-component:** $-3 + 8 - 2 = 3$\n- **$\\mathbf{k}$-component:** $5 + 1 - 4 = 2$\n\nShowing the step of summing these coefficients earns the second method mark **(M1)**.\n\nCombining these results gives the resultant force vector:\n\n$$\\mathbf{R} = 4\\mathbf{i} + 3\\mathbf{j} + 2\\mathbf{k}$$\n\nThis final exact expression completes the steps to secure the accuracy mark **(A1)**."}]},{"id":"column_vector","label":"Column Vector Addition","steps":[{"type":"text","order":0,"title":"Define the resultant force","content":"To find the resultant force, we need to find the sum of all individual forces acting on the boat. \n\n$$\\mathbf{F}_{\\text{resultant}} = \\mathbf{F}_1 + \\mathbf{F}_2 + \\mathbf{F}_3$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\mathbf{F}_1 = 2\\mathbf{i} - 3\\mathbf{j} + 5\\mathbf{k} \\implies \\htmlData{anchor=f1-col}{\\begin{pmatrix} 2 \\\\ -3 \\\\ 5 \\end{pmatrix}}$"},{"latex":"$$\\mathbf{F}_2 = -4\\mathbf{i} + 8\\mathbf{j} + \\mathbf{k} \\implies \\htmlData{anchor=f2-col}{\\begin{pmatrix} -4 \\\\ 8 \\\\ 1 \\end{pmatrix}}$"},{"latex":"$$\\mathbf{F}_3 = 6\\mathbf{i} - 2\\mathbf{j} - 4\\mathbf{k} \\implies \\htmlData{anchor=f3-col}{\\begin{pmatrix} 6 \\\\ -2 \\\\ -4 \\end{pmatrix}}$"}],"highlights":[{"color":"sky","style":"box","anchor":"f1-col"},{"color":"amber","style":"box","anchor":"f2-col"},{"color":"purple","style":"box","anchor":"f3-col"}],"connections":[]},"order":1,"title":"Convert to column vectors","description":"We convert each of the given force vectors from unit vector form (using $\\mathbf{i}$, $\\mathbf{j}$, and $\\mathbf{k}$) into column vector form. This representation makes it easier to perform component-wise arithmetic."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\mathbf{F}_{\\text{resultant}} = \\htmlData{anchor=sum-expr}{\\begin{pmatrix} 2 \\\\ -3 \\\\ 5 \\end{pmatrix} + \\begin{pmatrix} -4 \\\\ 8 \\\\ 1 \\end{pmatrix} + \\begin{pmatrix} 6 \\\\ -2 \\\\ -4 \\end{pmatrix}}$"},{"latex":"$$\\phantom{\\mathbf{F}_{\\text{resultant}}} = \\begin{pmatrix} 2 + (-4) + 6 \\\\ -3 + 8 + (-2) \\\\ 5 + 1 + (-4) \\end{pmatrix}$"},{"latex":"$$\\phantom{\\mathbf{F}_{\\text{resultant}}} = \\htmlData{anchor=sum-res}{\\begin{pmatrix} 4 \\\\ 3 \\\\ 2 \\end{pmatrix}}$"}],"highlights":[{"color":"teal","style":"box","anchor":"sum-expr"},{"color":"teal","style":"box","anchor":"sum-res"}],"connections":[{"to":"sum-res","from":"sum-expr","color":"teal","label":"sum components","style":"solid"}]},"order":2,"title":"Add the column vectors","description":"Now, add the components along each axis (top row for $\\mathbf{i}$, middle row for $\\mathbf{j}$, and bottom row for $\\mathbf{k}$):\n\n- **$x$-component:** $2 - 4 + 6 = 4$\n- **$y$-component:** $-3 + 8 - 2 = 3$\n- **$z$-component:** $5 + 1 - 4 = 2$"},{"type":"text","order":3,"title":"Convert back to unit form","content":"Finally, convert the resulting column vector back into unit vector form to state the final answer:\n\n$$\\mathbf{F}_{\\text{resultant}} = 4\\mathbf{i} + 3\\mathbf{j} + 2\\mathbf{k}$$"}]}],"generatedAt":"2026-06-09T08:33:37.989Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"21697987-414a-4ad9-87e4-4de802ac1176:unit_vector","label":"Unit Vector Addition","steps":[{"type":"text","order":0,"title":"Understand the goal","content":"To find the resultant force acting on the boat, we need to sum the three individual force vectors. The formula is:\n\n$$\\mathbf{R} = \\mathbf{F}_1 + \\mathbf{F}_2 + \\mathbf{F}_3$$\n\nIdentifying that the resultant force is found by summing the individual forces earns the first method mark **(M1)**.","highlights":[{"text":"Find the resultant force.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\mathbf{F}_1 = \\htmlData{anchor=f1-i}{\\color{@sky}{2}}\\mathbf{i} \\htmlData{anchor=f1-j}{\\color{@amber}{- 3}}\\mathbf{j} \\htmlData{anchor=f1-k}{\\color{@purple}{+ 5}}\\mathbf{k}$"},{"latex":"$$\\mathbf{F}_2 = \\htmlData{anchor=f2-i}{\\color{@sky}{-4}}\\mathbf{i} \\htmlData{anchor=f2-j}{\\color{@amber}{+ 8}}\\mathbf{j} \\htmlData{anchor=f2-k}{\\color{@purple}{+ 1}}\\mathbf{k}$"},{"latex":"$$\\mathbf{F}_3 = \\htmlData{anchor=f3-i}{\\color{@sky}{+6}}\\mathbf{i} \\htmlData{anchor=f3-j}{\\color{@amber}{- 2}}\\mathbf{j} \\htmlData{anchor=f3-k}{\\color{@purple}{- 4}}\\mathbf{k}$"},{"latex":"$$\\mathbf{R} = (\\htmlData{anchor=sum-i}{\\color{@sky}{2 - 4 + 6}})\\mathbf{i} + (\\htmlData{anchor=sum-j}{\\color{@amber}{-3 + 8 - 2}})\\mathbf{j} + (\\htmlData{anchor=sum-k}{\\color{@purple}{5 + 1 - 4}})\\mathbf{k}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"f1-i"},{"color":"sky","style":"text-color","anchor":"f2-i"},{"color":"sky","style":"text-color","anchor":"f3-i"},{"color":"amber","style":"text-color","anchor":"f1-j"},{"color":"amber","style":"text-color","anchor":"f2-j"},{"color":"amber","style":"text-color","anchor":"f3-j"},{"color":"purple","style":"text-color","anchor":"f1-k"},{"color":"purple","style":"text-color","anchor":"f2-k"},{"color":"purple","style":"text-color","anchor":"f3-k"}],"connections":[{"to":"sum-i","from":"f1-i","color":"sky","label":null,"style":"dashed"},{"to":"sum-j","from":"f1-j","color":"amber","label":null,"style":"dashed"},{"to":"sum-k","from":"f1-k","color":"purple","label":null,"style":"dashed"}]},"order":1,"title":"Group the components","description":"We can directly sum the corresponding coefficients of the $\\mathbf{i}$, $\\mathbf{j}$, and $\\mathbf{k}$ components."},{"type":"text","order":2,"title":"Calculate final vector","content":"Now we simplify the calculation for each component:\n\n- **$\\mathbf{i}$-component:** $2 - 4 + 6 = 4$\n- **$\\mathbf{j}$-component:** $-3 + 8 - 2 = 3$\n- **$\\mathbf{k}$-component:** $5 + 1 - 4 = 2$\n\nShowing the step of summing these coefficients earns the second method mark **(M1)**.\n\nCombining these results gives the resultant force vector:\n\n$$\\mathbf{R} = 4\\mathbf{i} + 3\\mathbf{j} + 2\\mathbf{k}$$\n\nThis final exact expression completes the steps to secure the accuracy mark **(A1)**."}]},{"id":"21697987-414a-4ad9-87e4-4de802ac1176:column_vector","label":"Column Vector Addition","steps":[{"type":"text","order":0,"title":"Define the resultant force","content":"To find the resultant force, we need to find the sum of all individual forces acting on the boat. \n\n$$\\mathbf{F}_{\\text{resultant}} = \\mathbf{F}_1 + \\mathbf{F}_2 + \\mathbf{F}_3$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\mathbf{F}_1 = 2\\mathbf{i} - 3\\mathbf{j} + 5\\mathbf{k} \\implies \\htmlData{anchor=f1-col}{\\begin{pmatrix} 2 \\\\ -3 \\\\ 5 \\end{pmatrix}}$"},{"latex":"$$\\mathbf{F}_2 = -4\\mathbf{i} + 8\\mathbf{j} + \\mathbf{k} \\implies \\htmlData{anchor=f2-col}{\\begin{pmatrix} -4 \\\\ 8 \\\\ 1 \\end{pmatrix}}$"},{"latex":"$$\\mathbf{F}_3 = 6\\mathbf{i} - 2\\mathbf{j} - 4\\mathbf{k} \\implies \\htmlData{anchor=f3-col}{\\begin{pmatrix} 6 \\\\ -2 \\\\ -4 \\end{pmatrix}}$"}],"highlights":[{"color":"sky","style":"box","anchor":"f1-col"},{"color":"amber","style":"box","anchor":"f2-col"},{"color":"purple","style":"box","anchor":"f3-col"}],"connections":[]},"order":1,"title":"Convert to column vectors","description":"We convert each of the given force vectors from unit vector form (using $\\mathbf{i}$, $\\mathbf{j}$, and $\\mathbf{k}$) into column vector form. This representation makes it easier to perform component-wise arithmetic."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\mathbf{F}_{\\text{resultant}} = \\htmlData{anchor=sum-expr}{\\begin{pmatrix} 2 \\\\ -3 \\\\ 5 \\end{pmatrix} + \\begin{pmatrix} -4 \\\\ 8 \\\\ 1 \\end{pmatrix} + \\begin{pmatrix} 6 \\\\ -2 \\\\ -4 \\end{pmatrix}}$"},{"latex":"$$\\phantom{\\mathbf{F}_{\\text{resultant}}} = \\begin{pmatrix} 2 + (-4) + 6 \\\\ -3 + 8 + (-2) \\\\ 5 + 1 + (-4) \\end{pmatrix}$"},{"latex":"$$\\phantom{\\mathbf{F}_{\\text{resultant}}} = \\htmlData{anchor=sum-res}{\\begin{pmatrix} 4 \\\\ 3 \\\\ 2 \\end{pmatrix}}$"}],"highlights":[{"color":"teal","style":"box","anchor":"sum-expr"},{"color":"teal","style":"box","anchor":"sum-res"}],"connections":[{"to":"sum-res","from":"sum-expr","color":"teal","label":"sum components","style":"solid"}]},"order":2,"title":"Add the column vectors","description":"Now, add the components along each axis (top row for $\\mathbf{i}$, middle row for $\\mathbf{j}$, and bottom row for $\\mathbf{k}$):\n\n- **$x$-component:** $2 - 4 + 6 = 4$\n- **$y$-component:** $-3 + 8 - 2 = 3$\n- **$z$-component:** $5 + 1 - 4 = 2$"},{"type":"text","order":3,"title":"Convert back to unit form","content":"Finally, convert the resulting column vector back into unit vector form to state the final answer:\n\n$$\\mathbf{F}_{\\text{resultant}} = 4\\mathbf{i} + 3\\mathbf{j} + 2\\mathbf{k}$$"}]}],"generatedAt":"2026-06-09T08:33:49.478Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"f7773d42-7c9c-4e8d-990e-f9611f1e9215","revisionId":"5db0324c-5aa6-422f-9402-f719527f3573","questionNumber":"2","questionType":"LA","specification":"Given that $|3\\mathbf{i} + k\\mathbf{j} + 2\\mathbf{k}| = 7$","difficulty":1,"difficultyBand":"easy","tier":null,"totalMarks":3,"sourceTitle":"12.2 Vectors in 3D","calculatorUse":null,"isCalculation":true,"parts":[{"id":"72509b87-91ce-4a5e-8499-7a67c8e8857e","label":"0","content":"Find the two possible values of $k$","markscheme":"- Sets up magnitude equation: $\\sqrt{3^2 + k^2 + 2^2} = 7$ M1\n- Squares both sides and simplifies: $13 + k^2 = 49$ M1\n- Solves for $k$: $k^2 = 36 \\Rightarrow k = \\pm 6$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Sets up the equation $\\sqrt{3^2 + k^2 + 2^2} = 7$.","code":"M"},{"id":"p2","marks":1,"description":"Simplifies to $k^2 = 36$.","code":"M"},{"id":"p3","marks":1,"description":"Identifies both possible values $k = 6$ and $k = -6$.","code":"A"}],"finalAnswers":["6, -6"],"generatedAt":"2026-05-06T03:35:06.500Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"magnitude_formula","label":"Standard Magnitude Formula","steps":[{"type":"text","order":0,"title":"Recall the magnitude formula","content":"To find the magnitude of a 3D vector $\\mathbf{v} = x\\mathbf{i} + y\\mathbf{j} + z\\mathbf{k}$, we use the standard 3D magnitude formula:\n\n$$|\\mathbf{v}| = \\sqrt{x^2 + y^2 + z^2}$$\n\nIn this problem, we are given that the magnitude of $3\\mathbf{i} + k\\mathbf{j} + 2\\mathbf{k}$ is $7$. We can set up our equation by substituting these components into the formula:\n\n$$\\sqrt{3^2 + k^2 + 2^2} = 7$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\sqrt{\\htmlData{anchor=left-side}{\\color{@sky}{3^2 + k^2 + 2^2}}} = \\htmlData{anchor=right-side}{\\color{@amber}{7}}$"},{"latex":"$$\\sqrt{\\htmlData{anchor=simplified-sum}{\\color{@sky}{13 + k^2}}} = 7$"},{"latex":"$$\\htmlData{anchor=squared-left}{\\color{@sky}{13 + k^2}} = \\htmlData{anchor=squared-right}{\\color{@amber}{49}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"left-side"},{"color":"amber","style":"text-color","anchor":"right-side"}],"connections":[{"to":"simplified-sum","from":"left-side","color":"sky","label":"Simplify 9 + k^2 + 4","style":"solid"},{"to":"squared-right","from":"right-side","color":"amber","label":"Square both sides","style":"solid"}]},"order":1,"title":"Simplify and square","description":"We simplify the expression inside the square root first:\n\n$$3^2 + k^2 + 2^2 = 9 + k^2 + 4 = 13 + k^2$$\n\nNow, square both sides of the equation to eliminate the square root, which gives the simplified equation."},{"type":"text","order":2,"title":"Solve for k","content":"Now, we isolate $k^2$ by subtracting $13$ from both sides:\n\n$$\\begin{aligned} k^2 &= 49 - 13 \\\\ k^2 &= 36 \\end{aligned}$$\n\nTo find the possible values of $k$, take the square root of both sides. Remember that taking the square root of a positive number yields both a positive and a negative solution:\n\n$$k = \\pm\\sqrt{36}$$\n$$k = \\pm 6$$\n\nThis gives our final answers: $k = 6$ or $k = -6$."}]},{"id":"dot_product","label":"Using Dot Product","steps":[{"type":"text","order":0,"title":"Recall the vector identity","content":"To find the magnitude of a vector without writing down a square root, we can use the fundamental dot product property relating a vector's dot product with itself to the square of its magnitude:\n\n$$\\mathbf{v} \\cdot \\mathbf{v} = |\\mathbf{v}|^2$$\n\nLet our vector be $\\mathbf{v} = 3\\mathbf{i} + k\\mathbf{j} + 2\\mathbf{k}$. Given that $|\\mathbf{v}| = 7$, we can write:\n\n$$\\mathbf{v} \\cdot \\mathbf{v} = 7^2$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\mathbf{v} \\cdot \\mathbf{v} = \\htmlData{anchor=comp-x}{\\color{@sky}{3^2}} + \\htmlData{anchor=comp-y}{\\color{@amber}{k^2}} + \\htmlData{anchor=comp-z}{\\color{@purple}{2^2}}$"},{"latex":"$$\\htmlData{anchor=sum-squares}{\\color{@sky}{9} + \\color{@amber}{k^2} + \\color{@purple}{4}} = \\htmlData{anchor=mag-sq}{\\color{@teal}{49}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"comp-x"},{"color":"amber","style":"text-color","anchor":"comp-y"},{"color":"purple","style":"text-color","anchor":"comp-z"},{"color":"teal","style":"text-color","anchor":"mag-sq"}],"connections":[{"to":"sum-squares","from":"comp-y","color":"amber","label":null,"style":"solid"}]},"order":1,"title":"Set up the equation","description":"Compute the dot product by squaring the individual $\\mathbf{i}$, $\\mathbf{j}$, and $\\mathbf{k}$ components and set the sum equal to $7^2 = 49$. This earns the first method mark (M1)."},{"type":"text","order":2,"title":"Simplify and solve","content":"Now, simplify the equation by combining the constant terms:\n\n$$13 + k^2 = 49 \\quad \\text{(M1)}$$\n\nSubtract $13$ from both sides to isolate $k^2$:\n\n$$k^2 = 36$$\n\nTaking the square root of both sides gives the two possible values for $k$:\n\n$$k = \\pm 6 \\quad \\text{(A1)}$$\n\nTherefore, the two possible values of $k$ are $6$ and $-6$."}]}],"generatedAt":"2026-06-09T10:21:28.911Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"72509b87-91ce-4a5e-8499-7a67c8e8857e:magnitude_formula","label":"Standard Magnitude Formula","steps":[{"type":"text","order":0,"title":"Recall the magnitude formula","content":"To find the magnitude of a 3D vector $\\mathbf{v} = x\\mathbf{i} + y\\mathbf{j} + z\\mathbf{k}$, we use the standard 3D magnitude formula:\n\n$$|\\mathbf{v}| = \\sqrt{x^2 + y^2 + z^2}$$\n\nIn this problem, we are given that the magnitude of $3\\mathbf{i} + k\\mathbf{j} + 2\\mathbf{k}$ is $7$. We can set up our equation by substituting these components into the formula:\n\n$$\\sqrt{3^2 + k^2 + 2^2} = 7$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\sqrt{\\htmlData{anchor=left-side}{\\color{@sky}{3^2 + k^2 + 2^2}}} = \\htmlData{anchor=right-side}{\\color{@amber}{7}}$"},{"latex":"$$\\sqrt{\\htmlData{anchor=simplified-sum}{\\color{@sky}{13 + k^2}}} = 7$"},{"latex":"$$\\htmlData{anchor=squared-left}{\\color{@sky}{13 + k^2}} = \\htmlData{anchor=squared-right}{\\color{@amber}{49}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"left-side"},{"color":"amber","style":"text-color","anchor":"right-side"}],"connections":[{"to":"simplified-sum","from":"left-side","color":"sky","label":"Simplify 9 + k^2 + 4","style":"solid"},{"to":"squared-right","from":"right-side","color":"amber","label":"Square both sides","style":"solid"}]},"order":1,"title":"Simplify and square","description":"We simplify the expression inside the square root first:\n\n$$3^2 + k^2 + 2^2 = 9 + k^2 + 4 = 13 + k^2$$\n\nNow, square both sides of the equation to eliminate the square root, which gives the simplified equation."},{"type":"text","order":2,"title":"Solve for k","content":"Now, we isolate $k^2$ by subtracting $13$ from both sides:\n\n$$\\begin{aligned} k^2 &= 49 - 13 \\\\ k^2 &= 36 \\end{aligned}$$\n\nTo find the possible values of $k$, take the square root of both sides. Remember that taking the square root of a positive number yields both a positive and a negative solution:\n\n$$k = \\pm\\sqrt{36}$$\n$$k = \\pm 6$$\n\nThis gives our final answers: $k = 6$ or $k = -6$."}]},{"id":"72509b87-91ce-4a5e-8499-7a67c8e8857e:dot_product","label":"Using Dot Product","steps":[{"type":"text","order":0,"title":"Recall the vector identity","content":"To find the magnitude of a vector without writing down a square root, we can use the fundamental dot product property relating a vector's dot product with itself to the square of its magnitude:\n\n$$\\mathbf{v} \\cdot \\mathbf{v} = |\\mathbf{v}|^2$$\n\nLet our vector be $\\mathbf{v} = 3\\mathbf{i} + k\\mathbf{j} + 2\\mathbf{k}$. Given that $|\\mathbf{v}| = 7$, we can write:\n\n$$\\mathbf{v} \\cdot \\mathbf{v} = 7^2$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\mathbf{v} \\cdot \\mathbf{v} = \\htmlData{anchor=comp-x}{\\color{@sky}{3^2}} + \\htmlData{anchor=comp-y}{\\color{@amber}{k^2}} + \\htmlData{anchor=comp-z}{\\color{@purple}{2^2}}$"},{"latex":"$$\\htmlData{anchor=sum-squares}{\\color{@sky}{9} + \\color{@amber}{k^2} + \\color{@purple}{4}} = \\htmlData{anchor=mag-sq}{\\color{@teal}{49}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"comp-x"},{"color":"amber","style":"text-color","anchor":"comp-y"},{"color":"purple","style":"text-color","anchor":"comp-z"},{"color":"teal","style":"text-color","anchor":"mag-sq"}],"connections":[{"to":"sum-squares","from":"comp-y","color":"amber","label":null,"style":"solid"}]},"order":1,"title":"Set up the equation","description":"Compute the dot product by squaring the individual $\\mathbf{i}$, $\\mathbf{j}$, and $\\mathbf{k}$ components and set the sum equal to $7^2 = 49$. This earns the first method mark (M1)."},{"type":"text","order":2,"title":"Simplify and solve","content":"Now, simplify the equation by combining the constant terms:\n\n$$13 + k^2 = 49 \\quad \\text{(M1)}$$\n\nSubtract $13$ from both sides to isolate $k^2$:\n\n$$k^2 = 36$$\n\nTaking the square root of both sides gives the two possible values for $k$:\n\n$$k = \\pm 6 \\quad \\text{(A1)}$$\n\nTherefore, the two possible values of $k$ are $6$ and $-6$."}]}],"generatedAt":"2026-06-11T03:08:51.867Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"147f0bd6-aa23-4e70-a063-bd829fab5713","revisionId":"94b33485-77a0-4a90-b964-882f005cae7d","questionNumber":null,"questionType":"LA","specification":"Three thrust vectors act on a drone $\\mathbf{F}_1 = 8\\mathbf{i} + \\mathbf{j} - 3\\mathbf{k}$, $\\mathbf{F}_2 = -3\\mathbf{i} - 5\\mathbf{j} + 4\\mathbf{k}$ and $\\mathbf{F}_3 = -2\\mathbf{i} + 7\\mathbf{j} + 2\\mathbf{k}$","difficulty":1,"difficultyBand":"easy","tier":null,"totalMarks":3,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"0a6c5155-9789-4f92-95da-c8519157e4c0","label":"a","content":"Find the resultant force.","markscheme":"- Resultant $= \\mathbf{F}_1 + \\mathbf{F}_2 + \\mathbf{F}_3$ M1\n- Resultant $= \\begin{pmatrix} 8-3-2 \\\\ 1-5+7 \\\\ -3+4+2 \\end{pmatrix} = \\begin{pmatrix} 3 \\\\ 3 \\\\ 3 \\end{pmatrix}$ M1\n- Resultant $= 3\\mathbf{i} + 3\\mathbf{j} + 3\\mathbf{k}$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"States or implies the resultant force is the sum of the three individual vectors.","code":"M"},{"id":"p2","marks":1,"description":"Correct calculation showing the addition of components, either as a column vector or sum of i, j, k terms.","code":"M"},{"id":"p3","marks":1,"description":"Correct final resultant force in vector notation.","code":"A"}],"finalAnswers":["3\\mathbf{i} + 3\\mathbf{j} + 3\\mathbf{k}"],"generatedAt":"2026-05-06T14:04:00.003Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"unit_vector_addition","label":"Direct Unit Vector Addition","steps":[{"type":"text","order":0,"title":"State the resultant force expression","content":"To find the resultant force, $\\mathbf{F}_{R}$, we need to sum the three individual thrust vectors acting on the drone:\n\n$$\\mathbf{F}_{R} = \\mathbf{F}_1 + \\mathbf{F}_2 + \\mathbf{F}_3$$\n\nThis is where we earn the first mark, **M1**.","highlights":[{"text":"Find the resultant force.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\mathbf{F}_1 = \\htmlData{anchor=f1_i}{\\color{@sky}{8\\mathbf{i}}} + \\htmlData{anchor=f1_j}{\\color{@amber}{1\\mathbf{j}}} - \\htmlData{anchor=f1_k}{\\color{@purple}{3\\mathbf{k}}}$"},{"latex":"$$\\mathbf{F}_2 = \\htmlData{anchor=f2_i}{\\color{@sky}{-3\\mathbf{i}}} - \\htmlData{anchor=f2_j}{\\color{@amber}{5\\mathbf{j}}} + \\htmlData{anchor=f2_k}{\\color{@purple}{4\\mathbf{k}}}$"},{"latex":"$$\\mathbf{F}_3 = \\htmlData{anchor=f3_i}{\\color{@sky}{-2\\mathbf{i}}} + \\htmlData{anchor=f3_j}{\\color{@amber}{7\\mathbf{j}}} + \\htmlData{anchor=f3_k}{\\color{@purple}{2\\mathbf{k}}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"f1_i"},{"color":"sky","style":"text-color","anchor":"f2_i"},{"color":"sky","style":"text-color","anchor":"f3_i"}],"connections":null},"order":1,"title":"Substitute the vector values","description":"Let's write down the sum with the full vector expressions, keeping track of the components in each direction: $\\mathbf{i}$, $\\mathbf{j}$, and $\\mathbf{k}$."},{"type":"text","order":2,"title":"Group the components","content":"Now we group the coefficients of $\\mathbf{i}$, $\\mathbf{j}$, and $\\mathbf{k}$ together:\n\n$$\\mathbf{F}_{R} = (8 - 3 - 2)\\mathbf{i} + (1 - 5 + 7)\\mathbf{j} + (-3 + 4 + 2)\\mathbf{k}$$\n\nThis grouping of components earns the second method mark, **M1**."},{"type":"text","order":3,"title":"Calculate the final resultant","content":"Simplify the arithmetic within each bracket to find the final coefficients:\n\n* For $\\mathbf{i}$: $8 - 3 - 2 = 3$\n* For $\\mathbf{j}$: $1 - 5 + 7 = 3$\n* For $\\mathbf{k}$: $-3 + 4 + 2 = 3$\n\nSubstituting these values back gives:\n\n$$\\mathbf{F}_{R} = 3\\mathbf{i} + 3\\mathbf{j} + 3\\mathbf{k}\\text{ N}$$\n\nThis final vector representation earns the accuracy mark, **A1**."}]},{"id":"column_vector_addition","label":"Column Vector Conversion","steps":[{"type":"text","order":0,"title":"Define the resultant force","content":"To find the resultant force $\\mathbf{F}_{\\text{net}}$ acting on the drone, we need to add the three individual force vectors together:\n\n$$\\mathbf{F}_{\\text{net}} = \\mathbf{F}_1 + \\mathbf{F}_2 + \\mathbf{F}_3$$\n\nThis principle of adding force vectors earns us the first method mark (**M1**)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\mathbf{F}_1 = 8\\mathbf{i} + 1\\mathbf{j} - 3\\mathbf{k} \\longrightarrow \\htmlData{anchor=f1-col}{\\begin{pmatrix} 8 \\\\ 1 \\\\ -3 \\end{pmatrix}}$"},{"latex":"$$\\mathbf{F}_2 = -3\\mathbf{i} - 5\\mathbf{j} + 4\\mathbf{k} \\longrightarrow \\htmlData{anchor=f2-col}{\\begin{pmatrix} -3 \\\\ -5 \\\\ 4 \\end{pmatrix}}$"},{"latex":"$$\\mathbf{F}_3 = -2\\mathbf{i} + 7\\mathbf{j} + 2\\mathbf{k} \\longrightarrow \\htmlData{anchor=f3-col}{\\begin{pmatrix} -2 \\\\ 7 \\\\ 2 \\end{pmatrix}}$"}],"highlights":[{"color":"sky","style":"box","anchor":"f1-col"},{"color":"amber","style":"box","anchor":"f2-col"},{"color":"purple","style":"box","anchor":"f3-col"}],"connections":null},"order":1,"title":"Convert to column vectors","description":"Using column vectors makes adding the $\\mathbf{i}$, $\\mathbf{j}$, and $\\mathbf{k}$ components straightforward and helps prevent sign errors."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\mathbf{F}_{\\text{net}} = \\htmlData{anchor=c1}{\\begin{pmatrix} 8 \\\\ 1 \\\\ -3 \\end{pmatrix}} + \\htmlData{anchor=c2}{\\begin{pmatrix} -3 \\\\ -5 \\\\ 4 \\end{pmatrix}} + \\htmlData{anchor=c3}{\\begin{pmatrix} -2 \\\\ 7 \\\\ 2 \\end{pmatrix}}$"},{"latex":"$$\\phantom{\\mathbf{F}_{\\text{net}}} = \\begin{pmatrix} \\htmlData{anchor=x-sum}{8 - 3 - 2} \\\\ \\htmlData{anchor=y-sum}{1 - 5 + 7} \\\\ \\htmlData{anchor=z-sum}{-3 + 4 + 2} \\end{pmatrix}$"},{"latex":"$$\\phantom{\\mathbf{F}_{\\text{net}}} = \\htmlData{anchor=res-col}{\\begin{pmatrix} 3 \\\\ 3 \\\\ 3 \\end{pmatrix}}$"}],"highlights":[{"color":"green","style":"box","anchor":"res-col"}],"connections":[{"to":"x-sum","from":"c1","color":"sky","label":"x-comp","style":"dashed"}]},"order":2,"title":"Sum the column vectors","description":"Now, we sum the corresponding components inside the vectors. This step earns the second method mark (**M1**)."},{"type":"text","order":3,"title":"Convert back to unit vector form","content":"Finally, we convert the resulting column vector back into $\\mathbf{i}$, $\\mathbf{j}$, $\\mathbf{k}$ unit vector form:\n\n$$\\begin{pmatrix} 3 \\\\ 3 \\\\ 3 \\end{pmatrix} = 3\\mathbf{i} + 3\\mathbf{j} + 3\\mathbf{k}$$\n\nThis gives us our final accuracy mark (**A1**).\n\n**Resultant Force:**\n$$3\\mathbf{i} + 3\\mathbf{j} + 3\\mathbf{k}$$"}]}],"generatedAt":"2026-06-09T03:30:19.552Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"0a6c5155-9789-4f92-95da-c8519157e4c0:unit_vector_addition","label":"Direct Unit Vector Addition","steps":[{"type":"text","order":0,"title":"State the resultant force expression","content":"To find the resultant force, $\\mathbf{F}_{R}$, we need to sum the three individual thrust vectors acting on the drone:\n\n$$\\mathbf{F}_{R} = \\mathbf{F}_1 + \\mathbf{F}_2 + \\mathbf{F}_3$$\n\nThis is where we earn the first mark, **M1**.","highlights":[{"text":"Find the resultant force.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\mathbf{F}_1 = \\htmlData{anchor=f1_i}{\\color{@sky}{8\\mathbf{i}}} + \\htmlData{anchor=f1_j}{\\color{@amber}{1\\mathbf{j}}} - \\htmlData{anchor=f1_k}{\\color{@purple}{3\\mathbf{k}}}$"},{"latex":"$$\\mathbf{F}_2 = \\htmlData{anchor=f2_i}{\\color{@sky}{-3\\mathbf{i}}} - \\htmlData{anchor=f2_j}{\\color{@amber}{5\\mathbf{j}}} + \\htmlData{anchor=f2_k}{\\color{@purple}{4\\mathbf{k}}}$"},{"latex":"$$\\mathbf{F}_3 = \\htmlData{anchor=f3_i}{\\color{@sky}{-2\\mathbf{i}}} + \\htmlData{anchor=f3_j}{\\color{@amber}{7\\mathbf{j}}} + \\htmlData{anchor=f3_k}{\\color{@purple}{2\\mathbf{k}}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"f1_i"},{"color":"sky","style":"text-color","anchor":"f2_i"},{"color":"sky","style":"text-color","anchor":"f3_i"}],"connections":null},"order":1,"title":"Substitute the vector values","description":"Let's write down the sum with the full vector expressions, keeping track of the components in each direction: $\\mathbf{i}$, $\\mathbf{j}$, and $\\mathbf{k}$."},{"type":"text","order":2,"title":"Group the components","content":"Now we group the coefficients of $\\mathbf{i}$, $\\mathbf{j}$, and $\\mathbf{k}$ together:\n\n$$\\mathbf{F}_{R} = (8 - 3 - 2)\\mathbf{i} + (1 - 5 + 7)\\mathbf{j} + (-3 + 4 + 2)\\mathbf{k}$$\n\nThis grouping of components earns the second method mark, **M1**."},{"type":"text","order":3,"title":"Calculate the final resultant","content":"Simplify the arithmetic within each bracket to find the final coefficients:\n\n* For $\\mathbf{i}$: $8 - 3 - 2 = 3$\n* For $\\mathbf{j}$: $1 - 5 + 7 = 3$\n* For $\\mathbf{k}$: $-3 + 4 + 2 = 3$\n\nSubstituting these values back gives:\n\n$$\\mathbf{F}_{R} = 3\\mathbf{i} + 3\\mathbf{j} + 3\\mathbf{k}\\text{ N}$$\n\nThis final vector representation earns the accuracy mark, **A1**."}]},{"id":"0a6c5155-9789-4f92-95da-c8519157e4c0:column_vector_addition","label":"Column Vector Conversion","steps":[{"type":"text","order":0,"title":"Define the resultant force","content":"To find the resultant force $\\mathbf{F}_{\\text{net}}$ acting on the drone, we need to add the three individual force vectors together:\n\n$$\\mathbf{F}_{\\text{net}} = \\mathbf{F}_1 + \\mathbf{F}_2 + \\mathbf{F}_3$$\n\nThis principle of adding force vectors earns us the first method mark (**M1**)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\mathbf{F}_1 = 8\\mathbf{i} + 1\\mathbf{j} - 3\\mathbf{k} \\longrightarrow \\htmlData{anchor=f1-col}{\\begin{pmatrix} 8 \\\\ 1 \\\\ -3 \\end{pmatrix}}$"},{"latex":"$$\\mathbf{F}_2 = -3\\mathbf{i} - 5\\mathbf{j} + 4\\mathbf{k} \\longrightarrow \\htmlData{anchor=f2-col}{\\begin{pmatrix} -3 \\\\ -5 \\\\ 4 \\end{pmatrix}}$"},{"latex":"$$\\mathbf{F}_3 = -2\\mathbf{i} + 7\\mathbf{j} + 2\\mathbf{k} \\longrightarrow \\htmlData{anchor=f3-col}{\\begin{pmatrix} -2 \\\\ 7 \\\\ 2 \\end{pmatrix}}$"}],"highlights":[{"color":"sky","style":"box","anchor":"f1-col"},{"color":"amber","style":"box","anchor":"f2-col"},{"color":"purple","style":"box","anchor":"f3-col"}],"connections":null},"order":1,"title":"Convert to column vectors","description":"Using column vectors makes adding the $\\mathbf{i}$, $\\mathbf{j}$, and $\\mathbf{k}$ components straightforward and helps prevent sign errors."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\mathbf{F}_{\\text{net}} = \\htmlData{anchor=c1}{\\begin{pmatrix} 8 \\\\ 1 \\\\ -3 \\end{pmatrix}} + \\htmlData{anchor=c2}{\\begin{pmatrix} -3 \\\\ -5 \\\\ 4 \\end{pmatrix}} + \\htmlData{anchor=c3}{\\begin{pmatrix} -2 \\\\ 7 \\\\ 2 \\end{pmatrix}}$"},{"latex":"$$\\phantom{\\mathbf{F}_{\\text{net}}} = \\begin{pmatrix} \\htmlData{anchor=x-sum}{8 - 3 - 2} \\\\ \\htmlData{anchor=y-sum}{1 - 5 + 7} \\\\ \\htmlData{anchor=z-sum}{-3 + 4 + 2} \\end{pmatrix}$"},{"latex":"$$\\phantom{\\mathbf{F}_{\\text{net}}} = \\htmlData{anchor=res-col}{\\begin{pmatrix} 3 \\\\ 3 \\\\ 3 \\end{pmatrix}}$"}],"highlights":[{"color":"green","style":"box","anchor":"res-col"}],"connections":[{"to":"x-sum","from":"c1","color":"sky","label":"x-comp","style":"dashed"}]},"order":2,"title":"Sum the column vectors","description":"Now, we sum the corresponding components inside the vectors. This step earns the second method mark (**M1**)."},{"type":"text","order":3,"title":"Convert back to unit vector form","content":"Finally, we convert the resulting column vector back into $\\mathbf{i}$, $\\mathbf{j}$, $\\mathbf{k}$ unit vector form:\n\n$$\\begin{pmatrix} 3 \\\\ 3 \\\\ 3 \\end{pmatrix} = 3\\mathbf{i} + 3\\mathbf{j} + 3\\mathbf{k}$$\n\nThis gives us our final accuracy mark (**A1**).\n\n**Resultant Force:**\n$$3\\mathbf{i} + 3\\mathbf{j} + 3\\mathbf{k}$$"}]}],"generatedAt":"2026-06-09T03:30:31.026Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null}],"subjectCode":"a_level_maths__aqa","summaries":[{"id":"1773e25c-6e55-47e5-88eb-17ce1a44fefd","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"2487ca65-519c-4d2a-a2d6-d7c8ce0b3566","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"353fc7a7-ffab-4fb1-a2b4-7deb25f231f3","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"73269be0-2f98-40a2-aa60-af60de159afd","difficultyBand":"hard","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"8c3ba1de-3a9a-4561-befa-dad9b4bfe754","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"a4a45fda-f8bf-4de8-bcd3-fe0361d1bec0","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"aa02c06e-61f4-4de6-8646-c9658834a5f2","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"ac4dfeb8-9294-495c-9a3a-e9f0ae6ab65f","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"b5665545-09fe-464d-8e22-6905b128ddc0","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"b7c9fd5c-7475-46c5-91cb-5f27b36e97b7","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"dd413996-2ef1-496c-8ad7-32a3f2ecf1ad","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"f6e2e09a-6b07-4381-b754-b4f93d6e85e9","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"f7773d42-7c9c-4e8d-990e-f9611f1e9215","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"147f0bd6-aa23-4e70-a063-bd829fab5713","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"3a257466-f33f-49b1-a5eb-bdc57a116eda","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"5a65ec5a-6ce6-46b2-9642-ffe58c917a1d","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"61f1cd09-0dfa-4e3a-87c7-c9cc4de7d88b","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"693cb2a8-3308-401a-8a66-d17c6ff502e0","difficultyBand":"hard","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"7080b011-f573-47c2-b80a-66319949a18f","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"ac34d1b1-26d1-453e-ae64-d8fba2b64674","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"ace5614e-93cc-4afb-8cdc-6527de6c4148","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"b639201a-908c-40cc-9d39-380001a0e364","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"d38221f3-45d0-40f0-9afb-1b4cc04f15fa","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"12d25d19-6430-49e8-be62-000a457c552c","difficultyBand":"hard","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"1ad197c4-ec2f-4e84-a0ab-96410490a37e","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"2a789d5d-107f-4097-ac02-1a8d8ea550e3","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"4f8b540b-67b3-41ae-a98a-def2ba84d7ee","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"56b3d5ac-f7f6-4bf4-9039-17ede987db97","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"9b33877b-defc-4084-b8eb-677e8e52ae86","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"bf0e1e6a-5734-404e-8617-bf9bde649501","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"d30cbcf6-db2c-4454-b8ac-51c5b1481433","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"e08b82e7-2561-491c-bc71-ce3f9bdda4f6","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"e73d7704-b974-4aeb-83a1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Vectors to solve problems","topicId":"64e562f7-03f1-4581-b887-72b333f310d1","topicSidebar":"$L3a","topicSlug":"vectors-to-solve-problems","topicTitle":"1.13.5 Vectors to solve problems"}]}],"$L3b","$L3c"]}]

1.13.5 Vectors to solve problems

Question 4

1.13.5 Vectors to solve problems Questions

1.13.5 Vectors to solve problems

Question 4

1.13.5 Vectors to solve problems Questions