:"solid"}]},"order":2,"title":"Perform the subtraction","description":"Now, substitute the column vectors into our equation $\\vec{RS} = \\vec{OS} - \\vec{OR}$ and subtract the corresponding components: $x$, $y$, and $z$."},{"type":"text","order":3,"title":"Convert back to unit vectors","content":"Finally, we convert our resulting column vector back into unit vector form using $\\mathbf{i}$, $\\mathbf{j}$, and $\\mathbf{k}$:\n\n$$\\vec{RS} = -3\\mathbf{i} - 4\\mathbf{j} + 3\\mathbf{k}$$\n\nThis confirms the correct vector components and completes the derivation."}]},{"id":"unit_vector","label":"Direct Unit Vector Subtraction","steps":[{"type":"text","order":0,"title":"Recall the vector relationship","content":"To find the vector $\\vec{RS}$ connecting point $R$ to point $S$, we find the difference between their position vectors relative to the origin $O$.\n\nThe relationship is given by:\n$$\\vec{RS} = \\vec{OS} - \\vec{OR}$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\vec{OS} = \\htmlData{anchor=os-vec}{\\color{@sky}{5\\mathbf{i} - 7\\mathbf{j} + \\mathbf{k}}}$"},{"latex":"$$\\vec{OR} = \\htmlData{anchor=or-vec}{\\color{@amber}{8\\mathbf{i} - 3\\mathbf{j} - 2\\mathbf{k}}}$"},{"latex":"$$\\vec{RS} = (\\htmlData{anchor=os-sub}{\\color{@sky}{5\\mathbf{i} - 7\\mathbf{j} + \\mathbf{k}}}) - (\\htmlData{anchor=or-sub}{\\color{@amber}{8\\mathbf{i} - 3\\mathbf{j} - 2\\mathbf{k}}})$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"os-vec"},{"color":"amber","style":"text-color","anchor":"or-vec"}],"connections":[{"to":"os-sub","from":"os-vec","color":"sky","label":"Substitute OS","style":"solid"},{"to":"or-sub","from":"or-vec","color":"amber","label":"Substitute OR","style":"solid"}]},"order":1,"title":"Set up subtraction","description":"We write the vector subtraction explicitly using the unit vectors $\\mathbf{i}$, $\\mathbf{j}$, and $\\mathbf{k}$."},{"type":"text","order":2,"title":"Subtract component by component","content":"Now we group and subtract the corresponding coefficients for each of the unit vectors $\\mathbf{i}$, $\\mathbf{j}$, and $\\mathbf{k}$:\n\n$$\\vec{RS} = (5 - 8)\\mathbf{i} + (-7 - (-3))\\mathbf{j} + (1 - (-2))\\mathbf{k}$$\n\nLet's calculate each component:\n- $\\mathbf{i}$-component: $5 - 8 = -3$\n- $\\mathbf{j}$-component: $-7 + 3 = -4$\n- $\\mathbf{k}$-component: $1 + 2 = 3$\n\nThis algebraic subtraction earns the first two marks on the exam (M1 A1)."},{"type":"text","order":3,"title":"State final vector","content":"Combining these components gives the final vector:\n\n$$\\vec{RS} = -3\\mathbf{i} - 4\\mathbf{j} + 3\\mathbf{k}$$\n\nThis final exact expression earns the last accuracy mark (A1)."}]}],"generatedAt":"2026-06-09T07:51:21.029Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"a83ab17e-2747-4d91-ae49-cc66cc057d95","label":"b","content":"Find $|\\vec{RS}|$","markscheme":"- Uses magnitude formula: $\\sqrt{(-3)^2 + (-4)^2 + 3^2}$ M1\n- Simplifies to $\\sqrt{9 + 16 + 9}$ M1\n- Final answer: $\\sqrt{34}$ A1","marks":3,"order":1,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Correct formula for the magnitude of a 3D vector.","code":"M"},{"id":"p2","marks":1,"description":"Correct calculation of the sum of squares.","code":"M"},{"id":"p3","marks":1,"description":"Correct final surd answer.","code":"A"}],"finalAnswers":["\\sqrt{34}"],"generatedAt":"2026-05-06T14:03:54.235Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"vector_subtraction","label":"Vector Subtraction and Magnitude","steps":[{"type":"hyper_latex","block":{"nodes":[{"text":"To find the vector $\\vec{RS}$, we use the vector subtraction method:"},{"latex":"$$$\\vec{RS} = \\vec{OS} - \\vec{OR}$$"},{"text":"Substitute the position vectors of $S$ and $R$ into the formula:"},{"latex":"$$$\\vec{RS} = \\htmlData{anchor=os-vec}{\\color{@sky}{(5\\mathbf{i} - 7\\mathbf{j} + \\mathbf{k})}} - \\htmlData{anchor=or-vec}{\\color{@amber}{(8\\mathbf{i}

1 Pure Mathematics

Question 109

1 Pure Mathematics Questions

1 Pure Mathematics

Question 109

1 Pure Mathematics Questions