2021 JPJC Promo Q9
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.
(i)
Find the exact length of projection of $\overrightarrow{PQ}$ onto ${{\prod }_{1}}$.
[3]
(ii)
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 $\left( -2,1,4 \right)$.
(iii)
Find the cartesian equation of ${{\prod }_{2}}$.
[2]
(iv)
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]
(v)
Hence or otherwise, find a vector equation of the line of intersection of ${{\prod }_{1}}$ and ${{\prod }_{2}}$.
[2]
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