The point $M$ has position vector relative to the origin $O$, given by $6\mathbf{i}-5\mathbf{j}+11\m

Question

The point $M$ has position vector relative to the origin $O$, given by $6\mathbf{i}-5\mathbf{j}+11\mathbf{k}$. The line ${{l}_{1}}$ has equation $x-7=\frac{y}{3}=\frac{z+2}{-2}$, and the plane $\pi $ has equation $4x-2y-z=30$. Show that ${{l}_{1}}$ lies in $\pi $. [2] Find a cartesian equation of the plane containing ${{l}_{1}}$ and $M$. [3] The point $N$ is the foot of perpendicular from $M$ to ${{l}_{1}}$. The line ${{l}_{2}}$ is the line passing through $M$ and $N$. Find the position vector of $N$ and the area of triangle $OMN$. [5] Find the acute angle between ${{l}_{2}}$ and $\pi $, giving your answer correct to the nearest $0.1{}^\circ $. [3]

Accepted Answer

(i) Shown; (ii) $-29x + 11y + 2z = -207$; (iii) $N = (4, -9, 4)$, Area = 44.2 units²; (iv) $\theta = 10.6°$

2019 VJC P1 Q10

2019 VJC P1 Q10

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