CAIE A-Level Mathematics 3.7.5 Parallel, Intersecting and Skew Lines

CAIE A-Level Mathematics 3.7.5 Parallel, Intersecting and Skew Lines
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise comparing direction vectors and equating components with independent parameters to decide whether 3D lines are parallel, intersecting or skew and find an intersection.

How this is tested

  • compare direction-vector ratios to determine whether the lines can be parallel
  • equate corresponding components using different parameters and solve two equations
  • check the remaining component to confirm an intersection or prove non-parallel lines are skew

Question 11(b)

[Maximum number: 4]

With respect to the origin O, the points A, B, C and D have position vectors given by

OA→=(153),OB→=(041),OC→=(1−31) and OD→=(3−54).\overrightarrow{O A}=\left(\begin{array}{l} 1 \\ 5 \\ 3 \end{array}\right), \quad \overrightarrow{O B}=\left(\begin{array}{l} 0 \\ 4 \\ 1 \end{array}\right), \quad \overrightarrow{O C}=\left(\begin{array}{r} 1 \\ -3 \\ 1 \end{array}\right) \quad \text { and } \quad \overrightarrow{O D}=\left(\begin{array}{r} 3 \\ -5 \\ 4 \end{array}\right) .

The line m passes through the points A and B.

Find the position vector of the point of intersection of m and the line passing through the points C and D.