Projection onto a plane and a reflected triangle · source reference

Question

Plane $\pi_1$ and line l are $\mathbf r\cdot(2,1,-2)=6$ and $(x-1)/3=(y+4)/2=z-5$ respectively. A has position vector $3\mathbf i-\mathbf j+4\mathbf k$.

(i) Find the shortest distance from A to $\pi_1$. [2]

(ii) B satisfies $\overrightarrow{AB}=-5\mathbf j-2\mathbf k$. Find the length of projection of AB onto $\pi_1$. [2]

(iii) Use (i),(ii) to find area ABC, where C is A reflected in $\pi_1$. [2]

(iv) Find the Cartesian equation of $\pi_2$ containing l and O. Hence find the intersection of $\pi_1,\pi_2$. [4]

[2010/AJC/Prelim/I/13]

Source working

(i) P$(3,0,0)$ lies on $\pi_1$ since $(3,0,0)\cdot(2,1,-2)=6$.

$$\overrightarrow{AP}=(3,0,0)-(3,-1,4)=(0,1,-4).$$ $$\text{distance}=\frac{|\overrightarrow{AP}\cdot(2,1,-2)|}{|(2,1,-2)|}=\frac1{\sqrt9}|(0,1,-4)\cdot(2,1,-2)|=\frac{|1+8|}3=9/3=3.$$

(ii)

$$\begin{aligned}\text{projection length}&=\frac{|\overrightarrow{AB}\times(2,1,-2)|}{|(2,1,-2)|}\\&=\tfrac13|(0,-5,-2)\times(2,1,-2)|\\&=\tfrac13|(12,-4,10)|=\tfrac23|(6,-2,5)|\\&=\tfrac23\sqrt{6^2+2^2+5^2}=\tfrac23\sqrt{65}.\end{aligned}$$

(iii)

$$\text{Area}(ABC)=\tfrac12 AC\cdot\tfrac23\sqrt{65}=\tfrac12(3\times2)(\tfrac23\sqrt{65})=2\sqrt{65}\text{ units}^2.$$

(iv) $l:\mathbf r=(1,-4,5)+\lambda(3,2,1)$, $\lambda\in\mathbb R$.

Two vectors parallel to $\pi_2$ are $(1,-4,5)$ and $(3,2,1)$; their product

$$(1,-4,5)\times(3,2,1)=(-14,14,14)=14(-1,1,1).$$

Plane through O: $\mathbf r\cdot(-1,1,1)=0$, or $-x+y+z=0$.

Solve $-x+y+z=0$ and $2x+y-2z=6$. GC gives $x=2+t,y=2,z=t$.

$$\mathbf r=(2,2,0)+\mu(1,0,1),\quad\mu\in\mathbb R.$$

Teacher notes

Point-to-plane distance: $|\overrightarrow{AP}\cdot\mathbf n|/|\mathbf n|=AP\cos\theta=AF$, n normal to $\pi_1$.

Length of projection of AB onto plane: $|\overrightarrow{AB}\times\mathbf n|/|\mathbf n|=AB\sin\alpha=B'F$, n normal.

GC steps: apps, 9↓PlySmlt2, 2: Simultaneous Eqn Solver, Eqns: 2, Unknowns: 3, NEXT, enter equations, SOLVE. $x_1=x,x_2=y,x_3=z$.

For a vector line equation, write $\mathbf r=\cdots$ and state the parameter belongs to $\mathbb R$.

Native diagrams

(i) P,F on plane, A above F, AP oblique, red AF with right angle, θ at A, normal $(2,1,-2)$. (ii) A$(3,-1,4)$, B, their projections F,B′, dashed perpendicular from B, angle α at A, red B′F and normal. (iii) A,C on opposite sides of plane, F midpoint with AF=FC=3, B, dashed perpendicular height to AC labelled $2\sqrt{65}/3$, and normal.

Printed answer key

(i) $3$ units; (ii) $2\sqrt{65}/3$; (iii) $2\sqrt{65}$ units squared; (iv) $-x+y+z=0$; intersection $\mathbf r=(2,2,0)+\mu(1,0,1)$, $\mu\in\mathbb R$.

Provenance

2025 TJC H2 Math (9758) JC2 MSM: VECTORS. Question document pages 3; solution document pages 15,16. Original page numbers are retained as text metadata.