Discussion questions · Section 2 of 2

$A$, $B$ and $C$ are points $(0,1,2)$, $(3,2,1)$ and $(1,-1,0)$ respectively.

(i) Find the unit vector perpendicular to both $\overrightarrow{AB}$ and $\overrightarrow{AC}$.

(ii) Find the cosine of the angles that $\overrightarrow{OB}$ makes with the x-, y- and z-axis.

Three non-zero vectors $\mathbf a$, $\mathbf b$ and $\mathbf c$ are such that $2\mathbf a\times\mathbf b=\mathbf a\times3\mathbf c$.

Find a linear relationship between $\mathbf a$, $\mathbf b$ and $\mathbf c$.

Given $|\mathbf a\times\mathbf b|=4$, find the value of $|(\mathbf a+2\mathbf b)\times(3\mathbf a-\mathbf b)|$.

The points $A$ and $B$ have position vectors $\mathbf a$ and $\mathbf b$ respectively, relative to the origin $O$ such that $|\mathbf a|=|\mathbf b|$. The point $P$ with position vector $\mathbf p$ lies on $AB$ such that $\mathbf b\cdot\mathbf p=\mathbf a\cdot\mathbf p$.

(i) Show that $\overrightarrow{AB}$ is perpendicular to $\overrightarrow{OP}$.

(ii) Determine the position vector of the point $D$ in terms of $\mathbf a$ and $\mathbf b$, where $D$ is the reflection of $O$ about the line $AB$.

(iii) Give the geometrical meaning of $|\mathbf a\times\mathbf b|$.

The points $A$, $B$, $C$ and $D$ have position vectors $\mathbf i-2\mathbf j+5\mathbf k$, $\mathbf i+3\mathbf j$, $10\mathbf i+\mathbf j+2\mathbf k$ and $-2\mathbf i+4\mathbf j+5\mathbf k$ respectively, with respect to an origin $O$. The point $P$ on $AB$ is such that $AP:PB=\lambda:1-\lambda$ and the point $Q$ on $CD$ is such that $CQ:QD=\mu:1-\mu$. Find $\overrightarrow{OP}$ and $\overrightarrow{OQ}$ in terms of $\lambda$ and $\mu$ respectively.

Given that $\overrightarrow{PQ}$ is perpendicular to both $\overrightarrow{AB}$ and $\overrightarrow{CD}$,

(i) show that $\overrightarrow{PQ}=\mathbf i+2\mathbf j+2\mathbf k$,

(ii) find the area of the triangle $ABQ$.

Relative to the origin $O$, three distinct fixed points $A$, $B$ and $C$ have position vectors $\mathbf a$, $\mathbf b$ and $\mathbf c$ respectively. It is known that $\mathbf b$ is a unit vector, $|\mathbf a|=3$, $|\mathbf c|=2$ and the angle $AOC$ is $60^\circ$.

(i) State the geometrical interpretation of $|\mathbf b\cdot\mathbf c|$.1 mark

It is further given that $\mathbf a\times2\mathbf b=\mathbf b\times\mathbf c$.

(ii) Find the ratio of the area of triangle $AOB$ to the area of triangle $BOC$.2 marks

(iii) Show that $2\mathbf a+\mathbf c=k\mathbf b$ where $k\in\mathbb R$, $k\ne0$.

By considering $(2\mathbf a+\mathbf c)\cdot(2\mathbf a+\mathbf c)$, find the exact values of $k$.6 marks