CAIE A-Level Further Math AS 1.6.1 Plane Equations Questions

Practise finding Cartesian equations of planes from points, vectors and lines.

Exam points

find Cartesian equations of planes
use vectors and points to determine plane equations
solve plane equation problems in context

[Maximum number: 4]

The lines $l_{1}$ and $l_{2}$ have equations

\mathbf{r}=-2 \mathbf{i}-3 \mathbf{j}-5 \mathbf{k}+\lambda(-4 \mathbf{i}+3 \mathbf{j}+5 \mathbf{k}) \quad \text { and } \quad \mathbf{r}=2 \mathbf{i}-2 \mathbf{j}+3 \mathbf{k}+\mu(2 \mathbf{i}-3 \mathbf{j}+\mathbf{k})

respectively.

Find an equation of $\Pi$ , giving your answer in the form a x+b y+c z=d.

$\left|\begin{array}{ccc}\mathbf{i} \mathbf{j} \mathbf{k} 1 0 -4 3 5\end{array}\right|=\left(\begin{array}{c}3 9 -3\end{array}\right) \sim\left(\begin{array}{c}1 3 -1\end{array}\right)$
Finds vector perpendicular to the plane.

$1(-1)+3(-3)-1(-4)=-6 \Rightarrow x+3 y-z=-6$
Uses point in the plane.

4

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CAIE A-Level Further Math AS 1.6.1 Plane Equations Questions question 1