CAIE A-Level Mathematics 3.7.4 Straight lines

CAIE A-Level Mathematics 3.7.4 Straight lines
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise forming and using vector equations of straight lines from points, direction vectors and point-on-line conditions.

How this is tested

  • write a line as r=a+tb using a point vector and a direction vector from two points
  • substitute a point into a line equation to show it lies on the line
  • use component equations from r=a+tb to find or verify parameters

Question 11(a)

[Maximum number: 2]

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 a vector equation for m.