A-Level Maths: Vectors

Vectors at A-Level extend from 2D arithmetic into 3D geometry, and the exam questions turn on a handful of precise ideas: magnitude, scalar multiples, position versus displacement, and the section formula. This quiz covers them all with concrete components throughout.

The foundational questions handle magnitude in 3D — the 3-4-12 vector whose length is exactly 13, and the distance of a point from the origin as the magnitude of its position vector, given in exact form √56 = 2√14. The parallel-vector questions test the defining criterion, scalar multiplication, twice: identifying which vectors are multiples of 2i − 6j + 4k (including the negative multiple −1.5v, the case students most often miss), and a statement-checking question on −i + 2j − 2k where the vector is parallel to i − 2j + 2k precisely because it equals −1 times it, alongside the magnitude and unit vector.

The position-vector questions cover the operational core: the displacement AB as b − a, the resultant 2a − b with its sign trap in the subtracted component, midpoints and exact distances in 3D from points P(1,2,3) and Q(4,5,6), and collinearity through scalar-multiple criteria. The section formula appears in ratio form: the point dividing AB in ratio 1:3, computed from the weighted average.

Kinematics closes the set with the constant-velocity position update r = vt — the vector bridge into mechanics.

Explanations show component-by-component working, the format that earns method marks on the written paper.

• Compute magnitudes and unit vectors in 2D and 3D
• Identify parallel vectors as scalar multiples, including negative multiples
• Form displacement vectors, resultants, midpoints and distances from position vectors
• Apply the section formula to divide a segment in a given ratio
• Use vectors in kinematics with constant velocity

Topic scope follows section J (Vectors) of the DfE's prescribed AS and A level mathematics subject content — 100% common across all exam boards: magnitude and direction, position vectors, 2D and (A level) 3D vectors, geometric problems, and applications to kinematics.