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7\\mathbf{j} + 4\\mathbf{k}| = 9$","difficulty":1,"difficultyBand":"easy","tier":null,"totalMarks":3,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"9e8a0366-e684-41b8-b884-0ddab9aa14cf","label":"a","content":"Find the two possible values of $k$","markscheme":"- Sets up magnitude equation: $\\sqrt{k^2 + 7^2 + 4^2} = 9$ or $k^2 + 49 + 16 = 81$ M1\n- Simplifies to $k^2 = C$: $k^2 + 65 = 81 \\Rightarrow k^2 = 16$ M1\n- Solves for $k$: $k = \\pm 4$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Applies magnitude formula: $k^2 + 7^2 + 4^2 = 9^2$.","code":"M"},{"id":"p2","marks":1,"description":"Isolates $k^2$ to find $k^2 = 16$.","code":"M"},{"id":"p3","marks":1,"description":"Correctly states both $k = 4$ and $k = -4$.","code":"A"}],"finalAnswers":["k = \\pm 4","k = 4, k = -4"],"generatedAt":"2026-05-06T14:03:43.750Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"algebraic","label":"Algebraic Formula Approach","steps":[{"type":"text","order":0,"title":"Recall 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 3D version of Pythagoras' theorem:\n\n$$|\\mathbf{v}| = \\sqrt{x^2 + y^2 + z^2}$$\n\nIn this question, we are given that the magnitude of the vector $k\\mathbf{i} + 7\\mathbf{j} + 4\\mathbf{k}$ is equal to $9$."},{"type":"text","order":1,"title":"Set up the equation","content":"Using the magnitude formula, we set the magnitude of the vector equal to $9$:\n\n$$\\sqrt{k^2 + 7^2 + 4^2} = 9$$\n\nTo make this equation easier to solve, we square both sides to eliminate the square root:\n\n$$k^2 + 7^2 + 4^2 = 9^2$$\n\n$$k^2 + 49 + 16 = 81$$\n\nThis step earns the first method mark (**M1**)."},{"type":"text","order":2,"title":"Isolate the variable term","content":"Now we simplify the equation by combining the constant numbers on the left-hand side:\n\n$$49 + 16 = 65$$\n\nThis simplifies our equation to:\n\n$$k^2 + 65 = 81$$\n\nNext, we subtract $65$ from both sides to isolate the $k^2$ term:\n\n$$k^2 = 81 - 65$$\n\n$$k^2 = 16$$\n\nThis simplified equation earns the second method mark (**M1**)."},{"type":"text","order":3,"title":"Find the solutions","content":"To solve for $k$, we 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$$k = \\pm \\sqrt{16}$$\n\n$$k = 4 \\quad \\text{or} \\quad k = -4$$\n\nThus, the two possible values of $k$ are $4$ and $-4$.\n\nThis final step earns the accuracy mark (**A1**)."},{"type":"text","order":4,"title":"Verify on calculator","content":"You can quickly verify your algebraic solution using a graphing calculator by plotting the functions and finding their intersection.","calculator":[{"gdc":{"expression":"y = \\sqrt{x^2 + 65}","instructions":"1. Open the Graph menu.\n2. Input f1(x) = \\sqrt{x^2 + 65} and f2(x) = 9.\n3. Graph both equations.\n4. Use the Intersection tool (G-Solve -> INTSECT or Analyze Graph -> Intersection) to find the x-values where the graphs cross.\n5. The intersections are at x = -4 and x = 4."},"ti84":{"expression":"Y_1 = \\sqrt{X^2 + 49 + 16}","instructions":"1. Press [Y=].\n2. Enter Y1 = \\sqrt{X^2 + 7^2 + 4^2} and Y2 = 9.\n3. Press [GRAPH] (adjust window to see X from -10 to 10 and Y from 0 to 15).\n4. Press [2nd] [CALC] and select 5: intersect.\n5. Follow the prompts to find the two intersection points at X = -4 and X = 4."},"desmos":{"expression":"y = \\sqrt{x^2 + 49 + 16}","instructions":"1. Type y = \\sqrt{x^2 + 49 + 16} in the first row.\n2. Type y = 9 in the second row.\n3. Click on the gray intersection points on the screen to reveal their coordinates: (-4, 9) and (4, 9)."},"description":"Graph the functions and find their intersection to verify the roots."}]}]}],"generatedAt":"2026-06-09T04:12:31.327Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"9e8a0366-e684-41b8-b884-0ddab9aa14cf:algebraic","label":"Algebraic Formula Approach","steps":[{"type":"text","order":0,"title":"Recall 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 3D version of Pythagoras' theorem:\n\n$$|\\mathbf{v}| = \\sqrt{x^2 + y^2 + z^2}$$\n\nIn this question, we are given that the magnitude of the vector $k\\mathbf{i} + 7\\mathbf{j} + 4\\mathbf{k}$ is equal to $9$."},{"type":"text","order":1,"title":"Set up the equation","content":"Using the magnitude formula, we set the magnitude of the vector equal to $9$:\n\n$$\\sqrt{k^2 + 7^2 + 4^2} = 9$$\n\nTo make this equation easier to solve, we square both sides to eliminate the square root:\n\n$$k^2 + 7^2 + 4^2 = 9^2$$\n\n$$k^2 + 49 + 16 = 81$$\n\nThis step earns the first method mark (**M1**)."},{"type":"text","order":2,"title":"Isolate the variable term","content":"Now we simplify the equation by combining the constant numbers on the left-hand side:\n\n$$49 + 16 = 65$$\n\nThis simplifies our equation to:\n\n$$k^2 + 65 = 81$$\n\nNext, we subtract $65$ from both sides to isolate the $k^2$ term:\n\n$$k^2 = 81 - 65$$\n\n$$k^2 = 16$$\n\nThis simplified equation earns the second method mark (**M1**)."},{"type":"text","order":3,"title":"Find the solutions","content":"To solve for $k$, we 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$$k = \\pm \\sqrt{16}$$\n\n$$k = 4 \\quad \\text{or} \\quad k = -4$$\n\nThus, the two possible values of $k$ are $4$ and $-4$.\n\nThis final step earns the accuracy mark (**A1**)."},{"type":"text","order":4,"title":"Verify on calculator","content":"You can quickly verify your algebraic solution using a graphing calculator by plotting the functions and finding their intersection.","calculator":[{"gdc":{"expression":"y = \\sqrt{x^2 + 65}","instructions":"1. Open the Graph menu.\n2. Input f1(x) = \\sqrt{x^2 + 65} and f2(x) = 9.\n3. Graph both equations.\n4. Use the Intersection tool (G-Solve -> INTSECT or Analyze Graph -> Intersection) to find the x-values where the graphs cross.\n5. The intersections are at x = -4 and x = 4."},"ti84":{"expression":"Y_1 = \\sqrt{X^2 + 49 + 16}","instructions":"1. Press [Y=].\n2. Enter Y1 = \\sqrt{X^2 + 7^2 + 4^2} and Y2 = 9.\n3. Press [GRAPH] (adjust window to see X from -10 to 10 and Y from 0 to 15).\n4. Press [2nd] [CALC] and select 5: intersect.\n5. Follow the prompts to find the two intersection points at X = -4 and X = 4."},"desmos":{"expression":"y = \\sqrt{x^2 + 49 + 16}","instructions":"1. Type y = \\sqrt{x^2 + 49 + 16} in the first row.\n2. Type y = 9 in the second row.\n3. Click on the gray intersection points on the screen to reveal their coordinates: (-4, 9) and (4, 9)."},"description":"Graph the functions and find their intersection to verify the roots."}]}]}],"generatedAt":"2026-06-09T04:12:52.018Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"4f8b540b-67b3-41ae-a98a-def2ba84d7ee","revisionId":"6c8b2856-d146-4998-a5fd-53fdcc038eb1","questionNumber":null,"questionType":"LA","specification":"Three forces act on a particle $\\mathbf{F}_1 = -10\\mathbf{i} + 3\\mathbf{j} + 2\\mathbf{k}$, $\\mathbf{F}_2 = 5\\mathbf{i} + 9\\mathbf{j} - 6\\mathbf{k}$ and $\\mathbf{F}_3 = 7\\mathbf{i} - 4\\mathbf{j} + 8\\mathbf{k}$","difficulty":1,"difficultyBand":"easy","tier":null,"totalMarks":3,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"df5d94ad-19b3-4fa3-9e5d-438efd50679c","label":"a","content":"Find the resultant force.","markscheme":"- Resultant $= \\mathbf{F}_1 + \\mathbf{F}_2 + \\mathbf{F}_3$ M1\n- Resultant $= \\begin{pmatrix} -10+5+7 \\\\ 3+9-4 \\\\ 2-6+8 \\end{pmatrix} = \\begin{pmatrix} 2 \\\\ 8 \\\\ 4 \\end{pmatrix}$ M1\n- Resultant $= 2\\mathbf{i} + 8\\mathbf{j} + 4\\mathbf{k}$ 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":["2\\mathbf{i} + 8\\mathbf{j} + 4\\mathbf{k}"],"generatedAt":"2026-05-06T14:04:02.212Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"unit_vectors","label":"Direct Unit Vector Addition","steps":[{"type":"text","order":0,"title":"Understand the method","content":"To find the resultant force of multiple forces acting on a particle, we sum the individual forces:\n\n$$\\mathbf{F}_{R} = \\mathbf{F}_1 + \\mathbf{F}_2 + \\mathbf{F}_3$$\n\nUsing the **Direct Unit Vector Addition** approach, we add the corresponding coefficients of the unit vectors $\\mathbf{i}$, $\\mathbf{j}$, and $\\mathbf{k}$ together.","highlights":[{"text":"Find the resultant force.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\mathbf{F}_1 = \\htmlData{anchor=i1}{\\color{@sky}{-10}}\\mathbf{i} + \\htmlData{anchor=j1}{\\color{@amber}{3}}\\mathbf{j} + \\htmlData{anchor=k1}{\\color{@purple}{2}}\\mathbf{k}$"},{"latex":"$$\\mathbf{F}_2 = \\htmlData{anchor=i2}{\\color{@sky}{5}}\\mathbf{i} + \\htmlData{anchor=j2}{\\color{@amber}{9}}\\mathbf{j} - \\htmlData{anchor=k2}{\\color{@purple}{6}}\\mathbf{k}$"},{"latex":"$$\\mathbf{F}_3 = \\htmlData{anchor=i3}{\\color{@sky}{7}}\\mathbf{i} - \\htmlData{anchor=j3}{\\color{@amber}{4}}\\mathbf{j} + \\htmlData{anchor=k3}{\\color{@purple}{8}}\\mathbf{k}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"i1"},{"color":"sky","style":"text-color","anchor":"i2"},{"color":"sky","style":"text-color","anchor":"i3"},{"color":"amber","style":"text-color","anchor":"j1"},{"color":"amber","style":"text-color","anchor":"j2"},{"color":"amber","style":"text-color","anchor":"j3"},{"color":"purple","style":"text-color","anchor":"k1"},{"color":"purple","style":"text-color","anchor":"k2"},{"color":"purple","style":"text-color","anchor":"k3"}],"connections":[]},"order":1,"title":"Set up the sum","description":"We identify the components of each force along the unit vectors $\\mathbf{i}$, $\\mathbf{j}$, and $\\mathbf{k}$."},{"type":"text","order":2,"title":"Calculate the components","content":"Now, we group and add the coefficients for each direction:\n\n- **$\\mathbf{i}$-component:** \n $$-10 + 5 + 7 = 2$$\n- **$\\mathbf{j}$-component:** \n $$3 + 9 - 4 = 8$$\n- **$\\mathbf{k}$-component:** \n $$2 - 6 + 8 = 4$$\n\nThis step earns the method marks **(M1, M1)** for correctly summing the components."},{"type":"text","order":3,"title":"State final resultant force","content":"Combining the calculated components back into unit vector form, the resultant force is:\n\n$$\\mathbf{F}_{R} = 2\\mathbf{i} + 8\\mathbf{j} + 4\\mathbf{k}$$\n\nThis awards the final accuracy mark **(A1)**."}]},{"id":"column_vectors","label":"Column Vector Representation","steps":[{"type":"text","order":0,"title":"Define the resultant force","content":"To find the resultant force $\\mathbf{F}_R$ acting on the particle, we need to find the vector sum of the three individual forces:\n\n$$\\mathbf{F}_R = \\mathbf{F}_1 + \\mathbf{F}_2 + \\mathbf{F}_3$$\n\nThis is our starting point and is worth the first method mark (**M1**).","highlights":[{"text":"Find the resultant force.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"text":"Using the Column Vector Representation approach, we write each force in the form:"},{"latex":"$$$\\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix}$$"},{"text":"Converting our given vectors:"},{"latex":"$$$\\mathbf{F}_1 = \\begin{pmatrix} \\htmlData{anchor=f1-i}{-10} \\\\ \\htmlData{anchor=f1-j}{3} \\\\ \\htmlData{anchor=f1-k}{2} \\end{pmatrix}, \\quad \\mathbf{F}_2 = \\begin{pmatrix} \\htmlData{anchor=f2-i}{5} \\\\ \\htmlData{anchor=f2-j}{9} \\\\ \\htmlData{anchor=f2-k}{-6} \\end{pmatrix}, \\quad \\mathbf{F}_3 = \\begin{pmatrix} \\htmlData{anchor=f3-i}{7} \\\\ \\htmlData{anchor=f3-j}{-4} \\\\ \\htmlData{anchor=f3-k}{8} \\end{pmatrix}$$"}],"highlights":[{"color":"sky","style":"box","anchor":"f1-i"},{"color":"sky","style":"box","anchor":"f2-i"},{"color":"sky","style":"box","anchor":"f3-i"}],"connections":[]},"order":1,"title":"Convert to column vectors","description":"We convert the $\\mathbf{i}$, $\\mathbf{j}$, and $\\mathbf{k}$ components of each force into column vector components."},{"type":"hyper_latex","block":{"nodes":[{"text":"Now, we add the corresponding components of the vectors together:"},{"latex":"$$$\\mathbf{F}_R = \\begin{pmatrix} \\htmlData{anchor=sum-i}{\\color{@sky}{-10} + \\color{@sky}{5} + \\color{@sky}{7}} \\\\ \\htmlData{anchor=sum-j}{\\color{@amber}{3} + \\color{@amber}{9} - \\color{@amber}{4}} \\\\ \\htmlData{anchor=sum-k}{\\color{@purple}{2} - \\color{@purple}{6} + \\color{@purple}{8}} \\end{pmatrix} = \\begin{pmatrix} \\htmlData{anchor=ans-i}{2} \\\\ \\htmlData{anchor=ans-j}{8} \\\\ \\htmlData{anchor=ans-k}{4} \\end{pmatrix}$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sum-i"},{"color":"amber","style":"text-color","anchor":"sum-j"},{"color":"purple","style":"text-color","anchor":"sum-k"}],"connections":[{"to":"ans-i","from":"sum-i","color":"sky","label":"x-component","style":"solid"},{"to":"ans-j","from":"sum-j","color":"amber","label":"y-component","style":"solid"},{"to":"ans-k","from":"sum-k","color":"purple","label":"z-component","style":"solid"}]},"order":2,"title":"Add the column vectors","description":"Adding the $x$, $y$, and $z$ components gives us the column vector for the resultant force. This calculation earns the second method mark (**M1**)."},{"type":"text","order":3,"title":"State the final vector","content":"Finally, we convert the column vector back into standard unit vector notation ($i, j, k$):\n\n$$\\mathbf{F}_R = 2\\mathbf{i} + 8\\mathbf{j} + 4\\mathbf{k}$$\n\nThis correct final expression awards the final accuracy mark (**A1**)."}]}],"generatedAt":"2026-06-09T02:00:18.171Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"df5d94ad-19b3-4fa3-9e5d-438efd50679c:unit_vectors","label":"Direct Unit Vector Addition","steps":[{"type":"text","order":0,"title":"Understand the method","content":"To find the resultant force of multiple forces acting on a particle, we sum the individual forces:\n\n$$\\mathbf{F}_{R} = \\mathbf{F}_1 + \\mathbf{F}_2 + \\mathbf{F}_3$$\n\nUsing the **Direct Unit Vector Addition** approach, we add the corresponding coefficients of the unit vectors $\\mathbf{i}$, $\\mathbf{j}$, and $\\mathbf{k}$ together.","highlights":[{"text":"Find the resultant force.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\mathbf{F}_1 = \\htmlData{anchor=i1}{\\color{@sky}{-10}}\\mathbf{i} + \\htmlData{anchor=j1}{\\color{@amber}{3}}\\mathbf{j} + \\htmlData{anchor=k1}{\\color{@purple}{2}}\\mathbf{k}$"},{"latex":"$$\\mathbf{F}_2 = \\htmlData{anchor=i2}{\\color{@sky}{5}}\\mathbf{i} + \\htmlData{anchor=j2}{\\color{@amber}{9}}\\mathbf{j} - \\htmlData{anchor=k2}{\\color{@purple}{6}}\\mathbf{k}$"},{"latex":"$$\\mathbf{F}_3 = \\htmlData{anchor=i3}{\\color{@sky}{7}}\\mathbf{i} - \\htmlData{anchor=j3}{\\color{@amber}{4}}\\mathbf{j} + \\htmlData{anchor=k3}{\\color{@purple}{8}}\\mathbf{k}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"i1"},{"color":"sky","style":"text-color","anchor":"i2"},{"color":"sky","style":"text-color","anchor":"i3"},{"color":"amber","style":"text-color","anchor":"j1"},{"color":"amber","style":"text-color","anchor":"j2"},{"color":"amber","style":"text-color","anchor":"j3"},{"color":"purple","style":"text-color","anchor":"k1"},{"color":"purple","style":"text-color","anchor":"k2"},{"color":"purple","style":"text-color","anchor":"k3"}],"connections":[]},"order":1,"title":"Set up the sum","description":"We identify the components of each force along the unit vectors $\\mathbf{i}$, $\\mathbf{j}$, and $\\mathbf{k}$."},{"type":"text","order":2,"title":"Calculate the components","content":"Now, we group and add the coefficients for each direction:\n\n- **$\\mathbf{i}$-component:** \n $$-10 + 5 + 7 = 2$$\n- **$\\mathbf{j}$-component:** \n $$3 + 9 - 4 = 8$$\n- **$\\mathbf{k}$-component:** \n $$2 - 6 + 8 = 4$$\n\nThis step earns the method marks **(M1, M1)** for correctly summing the components."},{"type":"text","order":3,"title":"State final resultant force","content":"Combining the calculated components back into unit vector form, the resultant force is:\n\n$$\\mathbf{F}_{R} = 2\\mathbf{i} + 8\\mathbf{j} + 4\\mathbf{k}$$\n\nThis awards the final accuracy mark **(A1)**."}]},{"id":"df5d94ad-19b3-4fa3-9e5d-438efd50679c:column_vectors","label":"Column Vector Representation","steps":[{"type":"text","order":0,"title":"Define the resultant force","content":"To find the resultant force $\\mathbf{F}_R$ acting on the particle, we need to find the vector sum of the three individual forces:\n\n$$\\mathbf{F}_R = \\mathbf{F}_1 + \\mathbf{F}_2 + \\mathbf{F}_3$$\n\nThis is our starting point and is worth the first method mark (**M1**).","highlights":[{"text":"Find the resultant force.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"text":"Using the Column Vector Representation approach, we write each force in the form:"},{"latex":"$$$\\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix}$$"},{"text":"Converting our given vectors:"},{"latex":"$$$\\mathbf{F}_1 = \\begin{pmatrix} \\htmlData{anchor=f1-i}{-10} \\\\ \\htmlData{anchor=f1-j}{3} \\\\ \\htmlData{anchor=f1-k}{2} \\end{pmatrix}, \\quad \\mathbf{F}_2 = \\begin{pmatrix} \\htmlData{anchor=f2-i}{5} \\\\ \\htmlData{anchor=f2-j}{9} \\\\ \\htmlData{anchor=f2-k}{-6} \\end{pmatrix}, \\quad \\mathbf{F}_3 = \\begin{pmatrix} \\htmlData{anchor=f3-i}{7} \\\\ \\htmlData{anchor=f3-j}{-4} \\\\ \\htmlData{anchor=f3-k}{8} \\end{pmatrix}$$"}],"highlights":[{"color":"sky","style":"box","anchor":"f1-i"},{"color":"sky","style":"box","anchor":"f2-i"},{"color":"sky","style":"box","anchor":"f3-i"}],"connections":[]},"order":1,"title":"Convert to column vectors","description":"We convert the $\\mathbf{i}$, $\\mathbf{j}$, and $\\mathbf{k}$ components of each force into column vector components."},{"type":"hyper_latex","block":{"nodes":[{"text":"Now, we add the corresponding components of the vectors together:"},{"latex":"$$$\\mathbf{F}_R = \\begin{pmatrix} \\htmlData{anchor=sum-i}{\\color{@sky}{-10} + \\color{@sky}{5} + \\color{@sky}{7}} \\\\ \\htmlData{anchor=sum-j}{\\color{@amber}{3} + \\color{@amber}{9} - \\color{@amber}{4}} \\\\ \\htmlData{anchor=sum-k}{\\color{@purple}{2} - \\color{@purple}{6} + \\color{@purple}{8}} \\end{pmatrix} = \\begin{pmatrix} \\htmlData{anchor=ans-i}{2} \\\\ \\htmlData{anchor=ans-j}{8} \\\\ \\htmlData{anchor=ans-k}{4} \\end{pmatrix}$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"sum-i"},{"color":"amber","style":"text-color","anchor":"sum-j"},{"color":"purple","style":"text-color","anchor":"sum-k"}],"connections":[{"to":"ans-i","from":"sum-i","color":"sky","label":"x-component","style":"solid"},{"to":"ans-j","from":"sum-j","color":"amber","label":"y-component","style":"solid"},{"to":"ans-k","from":"sum-k","color":"purple","label":"z-component","style":"solid"}]},"order":2,"title":"Add the column vectors","description":"Adding the $x$, $y$, and $z$ components gives us the column vector for the resultant force. This calculation earns the second method mark (**M1**)."},{"type":"text","order":3,"title":"State the final vector","content":"Finally, we convert the column vector back into standard unit vector notation ($i, j, k$):\n\n$$\\mathbf{F}_R = 2\\mathbf{i} + 8\\mathbf{j} + 4\\mathbf{k}$$\n\nThis correct final expression awards the final accuracy mark (**A1**)."}]}],"generatedAt":"2026-06-09T02:00:29.475Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null}],"subjectCode":"a_level_maths__ocr_mei","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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A","isCalculation":true},{"id":"f5605b6b-2aad-4052-85f1-5caa28656589","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true}],"title":"1.11.6 Vectors to solve problems","topicId":"300de5f2-89a7-441c-8f65-431bbc914d2a","topicSidebar":"$L3a","topicSlug":"vectors-to-solve-problems","topicTitle":"1.11.6 Vectors to solve problems"}]}],"$L3b","$L3c"]}]

1.11.6 Vectors to solve problems

Question 8

1.11.6 Vectors to solve problems Questions

1.11.6 Vectors to solve problems

Question 8

1.11.6 Vectors to solve problems Questions