With respect to the origin $O$, the position vectors of the points $A$ and $B$ are \(\mathbf{i

Question

With respect to the origin $O$, the position vectors of the points $A$ and $B$ are $\mathbf{i}+2\mathbf{j}+\mathbf{k}$ and $\mathbf{i}-4\mathbf{k}$ respectively. The equation of the planes ${{p}_{1}}$ and ${{p}_{2}}$ are given by ${{p}_{1}}:\mathbf{r}\cdot \left( \begin{matrix} 1 \ 3 \ 0 \ \end{matrix} \right)=1$ and ${{p}_{2}}:\mathbf{r}\cdot \left( \begin{matrix} 0 \ 7 \ \alpha \ \end{matrix} \right)=-4\alpha $, where $\alpha $ is a real constant with $\alpha >0$. Find the position vector of the foot of perpendicular from $A$ to $p_1$.[3] Verify that $B$ lies on both $p_1$ and $p_2$. Show that the vector equation of the line of intersection, $L$, of $p_1$ and $p_2$ is of the form $\mathbf{r}=\left( \begin{matrix} 1 \ 0 \ -4 \ \end{matrix} \right)+s\left( \begin{matrix} 3\alpha \ -\alpha \ 7 \ \end{matrix} \right)$, $s\in \mathbb{R}$. [2] The point $C(-1, 1, -4)$ is equidistant from both $p_1$ and $p_2$. Calculate the value of $\alpha$ for this case. Hence find in scalar product form the equation of the plane $p_3$ in which both the line $L$ and the point $C$ lie.[5] The plane $p_4$ has equation $y - z = 2$. Given that the angle between $L$ and $p_4$ is $\frac{\pi}{6}$, find the value of $\alpha$.[3]

Accepted Answer

(i) $\overrightarrow{OF}=\left( \begin{matrix} \frac{2}{5} \ \frac{1}{5} \ 1 \ \end{matrix} ight)$ (iii)(a) $\mathbf{r}\cdot \left( \begin{matrix} 1 \ 2 \ -3 \ \end{matrix} ight)=13$ (b) $\frac{7}{4}+\sqrt{\frac{147}{16}}$ or $4.78$ (since $\alpha >0$)

2023 ASRJC Promo Q11

2023 ASRJC Promo Q11