Category Archives: Vectors [A Level]


The equations of the plane π, and the lines L1 and l2 are given by

π: tx-2y+z=13

L1: r = (2i + j + 3k) + λ [2t i + (t² – 1) j – 2k]

L2: x – 4 = y, z = 5

where t and λ are real constants.

(a) Given that the shortest distance from the point P with coordinates (3, 3, -5) to π is 6, find the possible values of t.

(b) Show that L1 is parallel to π, and find a condition on t such that L1 is not on π.

For the case where t=2,

(i) show that L2 lies on π

(ii) given that A is a point on L1, and B is a point on L2, find the position vectors of A and B such that AB is perpendicular to both L1 and L2.

(iii) find the vector equation of the line of reflection of L1 in π.

Vectors – Artist’s Sculpture

[Source of question: Unknown]

As part of a sculpture, an artist erects a flat triangular sheet ABC in his garden. The vertices are attached to vertical poles DA, EB and FC. The coordinate axes Ox and Oy are horizontal, and Oz is vertical. The coordinates of triangle are ,  and , with unites in metres.

  1. Find the length of the side AC.
  2. Find the scalar product of AB⋅BC, and the angle BAC.
  3. Show 2i+3j+8k that is perpendicular to the lines AB and AC. Hence find the Cartesian equation of the plane ABC.
  4. The artist decides to erect another vertical pole GH based at the point G(1,1,0). Calculate the height of the pole if H is to lie in the plane ABC.


Handwritten Solution