A plane ${{\prod }_{1}}$ has equation $x+y+z=3$. A line passes through the points $P$ and \(Q\

Question

A plane ${{\prod }_{1}}$ has equation $x+y+z=3$. A line passes through the points $P$ and $Q$ with position vectors $\mathbf{i}+\mathbf{j}+2\mathbf{k}$ and $4\mathbf{i}+2\mathbf{j}-\mathbf{k}$ respectively. Find the exact length of projection of $\overrightarrow{PQ}$ onto ${{\prod }_{1}}$.[3] Find the position vector of the point of intersection of line $PQ$ and ${{\prod }_{1}}$.[3] A plane ${{\prod }_{2}}$ is parallel to the $y-z$ plane and contains the point $\left( -2,1,4 \right)$. Find the cartesian equation of ${{\prod }_{2}}$.[2] A point $S\left( a,7,b \right)$ lies on both ${{\prod }_{1}}$ and ${{\prod }_{2}}$. Write down the values of $a$ and $b$.[2] Hence or otherwise, find a vector equation of the line of intersection of ${{\prod }_{1}}$ and ${{\prod }_{2}}$.[2]

Accepted Answer

(i) $\displaystyle \frac{2\sqrt{42}}{3}\text{ units}$ (ii) $\displaystyle \overrightarrow{OA}=\begin{pmatrix}-2\ 0\ 5\end{pmatrix}$ (iii) $x=-2$ (iv) $a=-2,\ b=-2$ (v) $\displaystyle \mathbf{r}=\begin{pmatrix}-2\ 7\ -2\end{pmatrix}+\lambda\begin{pmatrix}0\ -1\ 1\end{pmatrix}$, $\lambda\in\mathbb{R}$

2021 JPJC Promo Q9

2021 JPJC Promo Q9

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