With reference to the origin $O$, points $A$ and $B$ have position vectors $\mathbf{a}$ and

Question

With reference to the origin $O$, points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively. Point $P$ lies on $AB$, such that $AP:PB=2:3$. Given that $OP$ and $AB$ are perpendicular, show that $\mathbf{a}\cdot\mathbf{b}=3|\mathbf{a}|^2-2|\mathbf{b}|^2$.[3] The plane $\pi$ has equation $\mathbf{r}=\begin{pmatrix}0\2\1\end{pmatrix}+\lambda\begin{pmatrix}-1\2\3\end{pmatrix}+\mu\begin{pmatrix}3\0\-1\end{pmatrix},\ \lambda,\mu\in\mathbb{R}.$ Show that a cartesian equation of $\pi$ is $x-4y+3z=-5$. [3] The line $l_1$ has equation $\mathbf{r}=\begin{pmatrix}6\-1\4\end{pmatrix}+t\begin{pmatrix}1\1\1\end{pmatrix},\ t\in\mathbb{R}.$ Find the acute angle between $l_1$ and $\pi$. [3] The line $l_2$ passes through the point $Q(-2,-1,2)$ and is parallel to $2\mathbf{i}+3\mathbf{j}+\mathbf{k}$. Find the position vector of the foot of the perpendicular from $Q$ to $\pi$. [3] Find the distance between $l_2$ and $\pi$. [2]

Accepted Answer

$\mathbf a\cdot\mathbf b=3|\mathbf a|^2-2|\mathbf b|^2$; $\pi:x-4y+3z=-5$; distance $27/\sqrt{26}$; $F=(-5/2,1,1/2)$; angle $21.5^\circ$

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2025 YIJC Promo Q8

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