CAIE A-Level Mathematics 3.7.6 Scalar Product

Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise calculating scalar products to find 3D angles, impose perpendicular conditions and locate the foot from a point to a line before deriving exact lengths or areas.

How this is tested

calculate a·b and divide by |a||b|, using an absolute value for an acute line angle
set the displacement to a general line point perpendicular to its direction vector
solve the resulting parameter and use the foot, cosine or side lengths for exact geometry

[Maximum number: 3]

The line $l_{1}$ passes through the point (3,1,-6) and is parallel to the vector 2 i+j+4 k.
The line $l_{2}$ passes through the point (-1,3,-6) and is perpendicular to the vector 3 i-2 j+k. The direction vector for $l_{2}$ has no component in the x-direction.

Calculate the acute angle between $l_{1}$ and $l_{2}$ .

How this is tested

Question 9(b)