With respect to an origin $O$, points $P$ and $Q$ have variable position vectors \(\mathbf{p}\

Question

With respect to an origin $O$, points $P$ and $Q$ have variable position vectors $\mathbf{p}$ and $\mathbf{q}$ respectively, given by $\mathbf{p}=(\cos t)\mathbf{i}+(\sin t)\mathbf{j}-\mathbf{k}$, $\mathbf{q}=(\cos 2t)\mathbf{i}-(\sin 2t)\mathbf{j}+\alpha \mathbf{k}$, where $\alpha$ is a real parameter and $0<t<\pi$. By considering the cross product or otherwise, find the exact values of $t$ and $\alpha$ if $O$, $P$ and $Q$ are collinear.[5] It is given that $\alpha=0.5$. Show that $\mathbf{p}\cdot \mathbf{q}=\cos 3t-0.5$. Find the greatest length of projection of $\mathbf{q}$ onto $\mathbf{p}$.[4]

Accepted Answer

(i)$t=\frac{\pi}{3}, a=1$ and $t=\frac{2\pi}{3}, a=-1$ (ii)$-\frac{3}{2\sqrt 2}$

2025 NJC P2 Q3

2025 NJC P2 Q3