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\n\nFinal answer: $-7\\mathbf{i} + 3\\mathbf{j}$ M1 A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Subtracts position vector $A$ from position vector $B$.","code":"M"},{"id":"p2","marks":1,"description":"Correct vector $-7\\mathbf{i} + 3\\mathbf{j}$ or $\\begin{pmatrix} -7 \\\\ 3 \\end{pmatrix}$.","code":"A"}],"finalAnswers":["-7i + 3j"],"generatedAt":"2026-05-12T02:14:06.603Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"position_vectors","label":"Subtract the position vectors","steps":[{"type":"text","order":0,"title":"Use end minus start","content":"To find a vector from $A$ to $B$, subtract the position vector of the starting point from the position vector of the ending point: $$\\vec{AB}=\\vec{OB}-\\vec{OA}.$$ This is the correct direction: end point minus start point.","highlights":[{"text":"Given that the point $A$ has position vector $3\\mathbf{i} + 4\\mathbf{j}$ and the point $B$ has position vector $-4\\mathbf{i} + 7\\mathbf{j}$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\vec{AB}=\\begin{pmatrix}\\color{@sky}{-4}\\\\\\color{@sky}{7}\\end{pmatrix}-\\begin{pmatrix}\\color{@amber}{3}\\\\\\color{@amber}{4}\\end{pmatrix}$"},{"latex":"$$\\vec{AB}=\\begin{pmatrix}\\color{@green}{-4-3}\\\\\\color{@green}{7-4}\\end{pmatrix}$"},{"latex":"$$\\vec{AB}=\\begin{pmatrix}\\color{@green}{-7}\\\\\\color{@green}{3}\\end{pmatrix}$"},{"latex":"$$\\vec{AB}=\\color{@green}{-7\\mathbf{i}+3\\mathbf{j}}$"},{"text":"The correct final vector earns the accuracy mark."}]},"order":1,"title":"Subtract components","description":"Write the position vectors as column vectors, then subtract corresponding components. The subtraction order is important: end point B minus start point A. Writing this numerical vector subtraction earns the method mark."}]}],"generatedAt":"2026-07-20T16:23:32.123Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"ea37e568-c264-47e2-abab-46317d143be4","label":"b","content":"Find $|\\overrightarrow{AB}|$","markscheme":"$$$\\sqrt{(-7)^2 + 3^2} = \\sqrt{49 + 9} = \\sqrt{58}$$ M1 A1","marks":2,"order":1,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Uses Pythagoras' theorem on their vector components.","code":"M"},{"id":"p2","marks":1,"description":"Final answer $\\sqrt{58}$.","code":"A"}],"finalAnswers":["\\sqrt{58}"],"generatedAt":"2026-05-12T02:14:06.603Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"component_magnitude","label":"Components then magnitude","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\vec{AB}=\\vec{OB}-\\vec{OA}$"},{"latex":"$$\\vec{AB}=\\htmlData{anchor=components}{\\color{@sky}{(-4-3)\\mathbf{i}+(7-4)\\mathbf{j}}}$"},{"latex":"$$\\vec{AB}=\\htmlData{anchor=result}{\\color{@green}{-7\\mathbf{i}+3\\mathbf{j}}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"components"},{"color":"green","style":"text-color","anchor":"result"}],"connections":[{"to":"result","from":"components","color":"sky","label":"simplify each component"}]},"order":0,"title":"Find the vector","description":"Subtract the position vector of A from the position vector of B, component by component."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$|\\vec{AB}|=\\sqrt{\\htmlData{anchor=squares}{\\color{@sky}{(-7)^2+3^2}}}$"},{"latex":"$$|\\vec{AB}|=\\sqrt{\\htmlData{anchor=sum}{\\color{@amber}{49+9}}}$"},{"latex":"$$|\\vec{AB}|=\\htmlData{anchor=answer}{\\color{@green}{\\sqrt{58}}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"squares"},{"color":"amber","style":"text-color","anchor":"sum"},{"color":"green","style":"text-color-with-background","anchor":"answer"}],"connections":[{"to":"sum","from":"squares","color":"sky","label":"evaluate the squares"}]},"order":1,"title":"Apply the magnitude","description":"Use Pythagoras on the horizontal and vertical components. Correct substitution earns M1, and the exact value √58 earns A1."}]}],"generatedAt":"2026-07-20T16:30:08.003Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":60},"walkthroughVideo":null},{"id":"77b099d4-1a73-4c36-821a-f6c132535fda","revisionId":"98d52c2a-1439-43fe-b356-eacc098e7446","questionNumber":"2","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","tier":null,"totalMarks":2,"sourceTitle":"Question Set 11: Vectors","calculatorUse":null,"isCalculation":true,"parts":[{"id":"107a09a5-52d2-4c28-b401-e3015fc5d6d3","label":"0","content":"Given that $|3\\mathbf{i} + k\\mathbf{j}| = 3\\sqrt{17}$\n\nFind the value of $k$","markscheme":"$$$\\sqrt{3^2 + k^2} = 3\\sqrt{17}$$\n$$3^2 + k^2 = (3\\sqrt{17})^2$$\n$$9 + k^2 = 153$$\n$$k^2 = 144$$\n$$k = \\pm 12$$ M1 A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Sets up equation $3^2 + k^2 = (3\\sqrt{17})^2$.","code":"M"},{"id":"p2","marks":1,"description":"Finds $k = \\pm 12$.","code":"A"}],"finalAnswers":["12","-12"],"generatedAt":"2026-05-12T02:14:06.603Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"magnitude_formula","label":"Use the magnitude formula","steps":[{"type":"text","order":0,"title":"Use the magnitude formula","content":"For a vector $a\\mathbf{i}+b\\mathbf{j}$, its magnitude is $\\sqrt{a^2+b^2}$. Here, $a=3$ and $b=k$, so substitute these components into the magnitude formula.","highlights":[{"text":"Given that $|3\\mathbf{i} + k\\mathbf{j}| = 3\\sqrt{17}$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=magnitude_start}{\\sqrt{3^2+k^2}}=\\htmlData{anchor=magnitude_rhs}{3\\sqrt{17}}$"},{"latex":"$$\\htmlData{anchor=squared_lhs}{3^2+k^2}=\\htmlData{anchor=squared_rhs}{(3\\sqrt{17})^2}$"},{"latex":"$$9+k^2=153$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"magnitude_start"},{"color":"sky","style":"text-color","anchor":"magnitude_rhs"},{"color":"orange","style":"text-color","anchor":"squared_rhs"}]},"order":1,"title":"Square both sides","description":"Square both sides to remove the square root, then simplify the numerical right-hand side. The complete magnitude equation earns the M1 method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=before_subtract}{9+k^2=153}$"},{"latex":"$$\\htmlData{anchor=after_subtract}{k^2=153-9}$"},{"latex":"$$k^2=144$"}],"highlights":[{"color":"green","style":"text-color","anchor":"before_subtract"},{"color":"green","style":"text-color","anchor":"after_subtract"}],"connections":[{"to":"after_subtract","from":"before_subtract","color":"green","label":"subtract 9","style":"solid"}]},"order":2,"title":"Isolate $k^2$","description":"Subtract 9 from both sides."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=root_start}{k^2=144}$"},{"latex":"$$\\htmlData{anchor=root_middle}{k=\\pm\\sqrt{144}}$"},{"latex":"$$\\htmlData{anchor=root_final}{k=\\pm12}$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"root_start"},{"color":"purple","style":"text-color","anchor":"root_middle"},{"color":"green","style":"text-color-with-background","anchor":"root_final"}],"connections":[{"to":"root_middle","from":"root_start","color":"purple","label":"take square roots"},{"to":"root_final","from":"root_middle","color":"green","label":"simplify"}]},"order":3,"title":"Take square roots","description":"Take square roots, remembering both signs because 12 and -12 both square to give 144. This earns the A1 mark."}]}],"generatedAt":"2026-07-20T16:27:47.549Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":44},"walkthroughVideo":null},{"id":"12e841aa-54a1-4f78-9229-64fa19db4936","revisionId":"c3a1f736-b739-4a83-9670-0a02db5cfa76","questionNumber":"3","questionType":"LA","specification":"Given that the point $A$ has position vector $-5\\mathbf{i} + 7\\mathbf{j}$ and the point $B$ has position vector $-8\\mathbf{i} + 2\\mathbf{j}$","difficulty":1,"difficultyBand":"easy","tier":null,"totalMarks":4,"sourceTitle":"Question Set 11: Vectors","calculatorUse":null,"isCalculation":true,"parts":[{"id":"c82c835c-c340-4ffb-89d6-1bb0d6a3fa72","label":"a","content":"Find the vector $\\vec{AB}$","markscheme":"$$$\\begin{pmatrix} -8 \\\\ 2 \\end{pmatrix} - \\begin{pmatrix} -5 \\\\ 7 \\end{pmatrix} = \\begin{pmatrix} -3 \\\\ -5 \\end{pmatrix}$$\n\nFinal answer: $-3\\mathbf{i} - 5\\mathbf{j}$ M1 A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Subtracts position vector $A$ from position vector $B$.","code":"M"},{"id":"p2","marks":1,"description":"Correct vector $-3\\mathbf{i} - 5\\mathbf{j}$.","code":"A"}],"finalAnswers":["-3i - 5j"],"generatedAt":"2026-05-12T02:14:06.603Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"position_vector_subtraction","label":"Subtract position vectors","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"\\vec{AB}=\\vec{OB}-\\vec{OA}"},{"latex":"\\vec{AB}=\\binom{\\color{@sky}{-8}}{\\color{@sky}{2}}-\\binom{\\color{@amber}{-5}}{\\color{@amber}{7}}"},{"latex":"\\vec{AB}=\\binom{\\color{@sky}{-8-(-5)}}{\\color{@sky}{2-7}}"},{"latex":"\\vec{AB}=\\binom{\\color{@green}{-3}}{\\color{@green}{-5}}"},{"latex":"\\boxed{\\vec{AB}=\\color{@green}{-3\\mathbf{i}-5\\mathbf{j}}}"}]},"order":0,"title":"Subtract the components","description":"Use the vector rule, then subtract the coordinates of A from the coordinates of B. Setting up B minus A earns M1. The fully simplified vector earns A1."}]}],"generatedAt":"2026-07-20T16:32:41.928Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"318e2eca-0d55-44a8-bafb-990166331bd3","label":"b","content":"Find $|\\vec{AB}|$","markscheme":"$$$\\sqrt{3^2 + 5^2} = \\sqrt{34}$$ M1 A1","marks":2,"order":1,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Uses Pythagoras' theorem on their vector components.","code":"M"},{"id":"p2","marks":1,"description":"Final answer $\\sqrt{34}$.","code":"A"}],"finalAnswers":["\\sqrt{34}"],"generatedAt":"2026-05-12T02:14:06.603Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"vector_magnitude","label":"Subtract then find magnitude","steps":[{"type":"text","order":0,"title":"Identify the coordinates","content":"Read the coefficients of $\\mathbf{i}$ and $\\mathbf{j}$ as coordinates:\n\n$$A=(-5,7),\\qquad B=(-8,2).$$\n\nTo find $\\vec{AB}$, subtract the coordinates of $A$ from those of $B$ because the vector travels **from $A$ to $B$**.","highlights":[{"text":"the point $A$ has position vector $-5\\mathbf{i} + 7\\mathbf{j}$ and the point $B$ has position vector $-8\\mathbf{i} + 2\\mathbf{j}$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\vec{AB}=\\color{@sky}{\\htmlData{anchor=subtract_rule}{B-A}}$"},{"latex":"$$\\vec{AB}=\\color{@sky}{\\htmlData{anchor=substitution}{\\bigl(-8-(-5)\\bigr)\\mathbf{i}+\\bigl(2-7\\bigr)\\mathbf{j}}}$"},{"latex":"$$\\vec{AB}=-3\\mathbf{i}-5\\mathbf{j}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"subtract_rule"},{"color":"sky","style":"text-color","anchor":"substitution"}]},"order":1,"title":"Find the displacement","description":"Subtract component by component. The negative signs show that the movement is left and down; they will disappear when the components are squared."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$|\\vec{AB}|=\\sqrt{\\htmlData{anchor=components}{\\color{@orange}{(-3)^2+(-5)^2}}}$"},{"latex":"$$|\\vec{AB}|=\\sqrt{9+25}$"},{"latex":"$$|\\vec{AB}|=\\htmlData{anchor=answer}{\\color{@green}{\\sqrt{34}}}$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"components"},{"color":"green","style":"text-color-with-background","anchor":"answer"}]},"order":2,"title":"Apply Pythagoras","description":"The magnitude is the length of the vector, so use Pythagoras on its horizontal and vertical components. Setting up the square-root calculation earns M1; obtaining the exact value square root of 34 earns 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