NYJC/2021/JC1/Promo/P1/Q09

The planes $\pi_1$ and $\pi_2$ have equations given by
$$\begin{array}{l} \pi_1: 4 x+3 y+5 z=7 \text { and } \ \pi_2: \mathbf{r}=2 \mathbf{i}+\mathbf{j}+\lambda(\mathbf{j}+\mathbf{k})+\mu(\mathbf{i}-\mathbf{j}-\mathbf{k}), \text { where } \lambda, \mu \in \mathbb{R} . \end{array}$$
(i) Find a vector equation of the line $h$ where $\pi_1$ and $\pi_2$ meet. Verify that point $P$ with position vector $-\mathbf{i}+2 \mathbf{j}+\mathbf{k}$ lies on $l_1$.

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(ii) The line $l_2$ which passes through $P$, lies in $\pi_2$ and is perpendicular to $l_1$. Find a cartesian equation of $l_2$.

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(iii) A point $Q(a, b, c)$ lies on $l_2$ where $a, b$ and $c$ are negative constants. Given that the distance from $Q$ to $\pi_1$ is $3 \sqrt{2}$, find the coordinates of $Q$. Hence or otherwise, find the exact length of projection of $\overrightarrow{Q P}$ on $\pi_1$

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