CAIE A-Level Mathematics 3.7.4 Straight lines

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

Exam points

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

[Maximum number: 3]

With respect to the origin O, the position vectors of the points A and B are given by $\overrightarrow{O A}=\left(\begin{array}{r}1 \\ 2 \\ -1\end{array}\right)$ and $\overrightarrow{O B}=\left(\begin{array}{l}0 \\ 3 \\ 1\end{array}\right)$ .

Find a vector equation for the line l through A and B.

Obtain direction vector -i+j+2 k, or equivalent
B1
Accept answers as column vectors throughout.
Use a correct method to form a vector equation
M1
State answer $\mathbf{r}=\mathbf{i}+2 \mathbf{j}-\mathbf{k}+\lambda(-\mathbf{i}+\mathbf{j}+2 \mathbf{k})$ , or equivalent correct form
A1
e.g. $\mathbf{r}=\left(\begin{array}{l}0 1\end{array}\right)+\mu\left(\begin{array}{c}1 -1 -2\end{array}\right)$ Allow $\left(\begin{array}{l}x y z\end{array}\right)$ for r.

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3.7.4—Straight lines question 1