The line $l$ contains the point $P$ with coordinates $\left( 1,-5,-5 \right)$ and is parallel

Question

The line $l$ contains the point $P$ with coordinates $\left( 1,-5,-5 \right)$ and is parallel to $3\mathbf{i}+\mathbf{j}-6\mathbf{k}$. The plane ${{\pi }_{1}}$ has cartesian equation $2x-5y-3z=4$. Find the acute angle between $l$ and ${{\pi }_{1}}$, giving your answer to the nearest degree.[3] Find the coordinates of $Q$, the point of intersection between $l$ and ${{\pi }_{1}}$.[3] The plane ${{\pi }_{2}}$ contains $l$ and is perpendicular to ${{\pi }_{1}}$. Find a cartesian equation of ${{\pi }_{2}}$.[3] Find a vector equation of the line where ${{\pi }_{1}}$ meets ${{\pi }_{2}}$.[2]

Accepted Answer

(i) $27^\circ$ (ii) $Q=(-5,-7,7)$ (iii) $33x+3y+17z=-67$ (iv) $\mathbf r=\begin{pmatrix}-5\-7\7\end{pmatrix}+t\begin{pmatrix}4\7\-9\end{pmatrix}$, where $t\in\mathbb R$

2024 JPJC Promo Q12

2024 JPJC Promo Q12

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