A parallel line and a perpendicular plane · source reference

Question

Line l passes through A$(-1,2,3)$ and B$(5,14,11)$. Plane p is $2x+3y-6z=-7$.

(i) Show l is parallel to p but not contained in it. [3]

(ii) Find the distance from l to p. [3]

(iii) Find a Cartesian equation of the plane containing l and perpendicular to p. [2]

[2010/JJC/Prelim/I/7]

Source working

(i) Direction $\overrightarrow{AB}=(5,14,11)-(-1,2,3)=(6,12,8)=2(3,6,4)$.

Plane p: $\mathbf r\cdot(2,3,-6)=-7$.

$$\mathbf d\cdot\mathbf n=(3,6,4)\cdot(2,3,-6)=6+18-24=0,$$

so l is parallel to p.

$$\overrightarrow{OA}\cdot\mathbf n=(-1,2,3)\cdot(2,3,-6)=-2+6-18=-14\ne-7.$$

A lies on l but not p, so l is parallel to but not contained in p.

(ii) C$(-7/2,0,0)$ lies on p since $(-7/2,0,0)\cdot(2,3,-6)=-7$.

$$\overrightarrow{CA}=(-1,2,3)-(-7/2,0,0)=(5/2,2,3).$$ $$\begin{aligned}\text{distance}&=\frac{|\overrightarrow{CA}\cdot(2,3,-6)|}{|(2,3,-6)|}\\&=\frac1{\sqrt{4+9+36}}|(5/2,2,3)\cdot(2,3,-6)|\\&=\frac{|5+6-18|}7=1\text{ unit}.\end{aligned}$$

(iii) Let required plane be $p_1$. A normal is $(3,6,4)\times(2,3,-6)=(-48,26,-3)$.

A lies on $p_1$, so

$$\mathbf r\cdot(-48,26,-3)=(-1,2,3)\cdot(-48,26,-3)=91,$$ $$-48x+26y-3z=91.$$

Teacher notes

$l\parallel p$ means the line direction is perpendicular to the plane normal: $\mathbf d\cdot\mathbf n=0$.

To show A does not lie on p, show its coordinates fail p’s equation. Pick any point C satisfying p’s equation to calculate distance.

A normal of $p_1$ is $\mathbf u\times\mathbf v$, where u,v are parallel to $p_1$: choose the normal of p and direction of l.

Native diagrams

Distance panel: plane p, C in it, l through A$(-1,2,3)$ above, a red perpendicular distance and normal $(2,3,-6)$. Perpendicular-plane panel: p and $p_1$ intersect, l lies in $p_1$, A on l, direction $(3,6,4)$ and normal $(2,3,-6)$ lie in $p_1$, with their right-angle relations.

Printed answer key

(ii) $1$ unit; (iii) $-48x+26y-3z=91$.

Provenance

2025 TJC H2 Math (9758) JC2 MSM: VECTORS. Question document pages 3; solution document pages 13,14. Original page numbers are retained as text metadata.