Vectors — AS & A Level Further Mathematics (9231) questions by topic
Vector equations of lines and planes fill this topic: finding a Cartesian equation ax + by + cz = d for a plane given in vector form, finding the shortest distance between two skew lines, and finding Cartesian equations of two planes both perpendicular to a given vector.
Find, show and express are the top command words across 16 questions from 2023 to 2025 papers, worth from 7 to 16 marks, where correctly identifying a normal vector via the cross or scalar triple product underpins nearly every part. With no examiner-report extracts logged for this topic, submitting each plane or line derivation for instant marking is the way to confirm a normal vector or shortest-distance formula has been applied correctly before the final equation is trusted.
May–June 2025, Paper 11
The points A, B, C have position vectors i - 2k, i + 2j + 2k, 2i - j - k, respectively. Find the equation of the plane ABC, giving your answer in the form ax + by + cz = d.
Answer this question and get it marked →May–June 2025, Paper 12
The points A, B, C have position vectors i - 2k, i + 2j + 2k, 2i - j - k, respectively. Find the equation of the plane ABC, giving your answer in the form ax + by + cz = d.
Answer this question and get it marked →May–June 2025, Paper 13
The plane Pi has equation r = 2i + 3j - 2k + lambda(i - 2j - k) + mu(3i + 2j - 2k). Find a Cartesian equation of Pi, giving your answer in the form ax + by + cz = d.
Answer this question and get it marked →May–June 2024, Paper 12
The points A, B, C have position vectors 2i + 2j + 4k, 2i + 4j - k, -3i - 3j + 4k, respectively, relative to the origin O. Find the equation of the plane ABC, giving your answer…
Answer this question and get it marked →May–June 2024, Paper 13
The lines l1 and l2 have equations r = i + 4j - k + lambda(j - 2k) and r = -3i + 4j + mu(i + 2j + k) respectively. Find the shortest distance between l1 and l2.
Answer this question and get it marked →May–June 2024, Paper 23
The planes Pi1 and Pi2 do not intersect and are both perpendicular to the vector i + 2j + 3k. The line l intersects Pi1 at the point (1, 6, 0) and intersects Pi2 at the point (3,…
Answer this question and get it marked →May–June 2024, Paper 33
A light spring of natural length a and modulus of elasticity kmg is attached to a fixed point O on a smooth plane inclined to the horizontal at an angle theta, where sin theta =…
Answer this question and get it marked →October–November 2024, Paper 11
The lines l1 and l2 have equations r = i + 3j - 2k + lambda(2i + j + k) and r = i - 2j + 9k + mu(i - 4j + 2k) respectively. The plane Pi1 contains l1 and is parallel to l2. Find…
Answer this question and get it marked →October–November 2024, Paper 12
The line l1 has equation r = i + 3j - k + lambda(i - j - 4k). The plane Pi contains l1 and is parallel to the vector 2i + 5j - 4k. Find the equation of Pi, giving your answer in…
Answer this question and get it marked →October–November 2024, Paper 13
The lines l1 and l2 have equations r = i + 3j - 2k + lambda(2i + j + k) and r = i - 2j + 9k + mu(i - 4j + 2k) respectively. The plane Pi1 contains l1 and is parallel to l2. Find…
Answer this question and get it marked →May–June 2023, Paper 11
The plane Pi1 has equation r = -4j - 3k + lambda(i - j + k) + mu(i + j - k). Obtain an equation of Pi1 in the form px + qy + rz = d.
Answer this question and get it marked →May–June 2023, Paper 12
The plane Pi1 has equation r = -4j - 3k + lambda(i - j + k) + mu(i + j - k). Obtain an equation of Pi1 in the form px + qy + rz = d.
Answer this question and get it marked →May–June 2023, Paper 13
The points A, B, C have position vectors i + j, -i + 2j + 4k, -2i + j + 3k, respectively, relative to the origin O. Find the equation of the plane ABC, giving your answer in the…
Answer this question and get it marked →October–November 2023, Paper 11
The lines l1 and l2 have equations r = -2i - 3j - 5k + lambda(-4i + 3j + 5k) and r = 2i - 2j + 3k + mu(2i - 3j + k) respectively. Find the shortest distance between l1 and l2.
Answer this question and get it marked →October–November 2023, Paper 12
The plane Pi1 has equation r = i - j - 2k + lambda(i - 2j - 3k) + mu(3i - k). Find an equation for Pi1 in the form ax + by + cz = d.
Answer this question and get it marked →October–November 2023, Paper 13
The lines l1 and l2 have equations r = -2i - 3j - 5k + lambda(-4i + 3j + 5k) and r = 2i - 2j + 3k + mu(2i - 3j + k) respectively. Find the shortest distance between l1 and l2.
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