The plane ${{\Pi }_{1}}$ and the line $~l$ have equations \({{\Pi }_{1}}:\mathbf{r}\cdot \left(

Question

The plane ${{\Pi }_{1}}$ and the line $~l$ have equations ${{\Pi }_{1}}:\mathbf{r}\cdot \left( \begin{matrix} 2 \ 1 \ -1 \ \end{matrix} \right)=3$ and $l:\frac{x+2}{3}=\frac{4-y}{2}=z-3$ respectively. Find the acute angle between ${{\Pi }_{1}}$ and $l$.[2] Find the coordinates of the point of intersection between ${{\Pi }_{1}}$ and $l$.[3] Find the perpendicular distance from $B\left( 10,\,-4,\,7 \right)$ to ${{\Pi }_{1}}$.[3] The plane ${{\Pi }_{2}}$ contains $l$ and is perpendicular to ${{\Pi }_{1}}$. Find a cartesian equation of ${{\Pi }_{2}}$.[3] Without using a calculator, find a vector equation of the line which lies in both ${{\Pi }_{1}}$ and ${{\Pi }_{2}}$.[2]

Accepted Answer

(a) $19.1^\circ$ (or $0.333$ radians) (b) $(4,0,5)$ (c) $\sqrt6$ units (d) $x+5y+7z=39$ (e) $\mathbf r=\begin{pmatrix}4\0\5\end{pmatrix}+\mu\begin{pmatrix}4\-5\3\end{pmatrix}$, $\mu\in\mathbb R$

2023 CJC P2 Q4

2023 CJC P2 Q4

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