2021 CJC Promo Q4
2021 CJC Promo Q4
8 marks
Promo
The points \(A\left( 1,0,-2 \right)\), \(B\left( 3,-1,-2 \right)\) and \(C\left( -3,7,0 \right)\) lie on plane \({{p}_{1}}\). Another plane \({{p}_{2}}\) has equation \(3x-y+2z=3\).
- Find a vector equation of plane \({{p}_{1}}\) in the form \(\mathbf{r}\cdot \mathbf{n}=d\).[3]
- Find the acute angle between \({{p}_{1}}\) and \({{p}_{2}}\).[2]
The equation of plane \({{p}_{3}}\) is given to be \(-9x+3y-6z=7\).
- Find the shortest distance between \({{p}_{2}}\) and \({{p}_{3}}\).[3]
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Answer:(i) \(\displaystyle \mathbf{r}\cdot\begin{pmatrix}1\\ 2\\ -5\end{pmatrix}=11\) (ii) \(\displaystyle \theta=\cos^{-1}\left(\frac{9}{\sqrt{420}}\right)=1.12\text{ rad}=64.0^\circ\) (iii) \(\displaystyle \frac{16}{3\sqrt{14}}\text{ units}=1.43\text{ units}\)
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