Practice section 6 of 9

Given $|\mathbf a|=4$, $|\mathbf b|=5$ and the angle between $\mathbf a$ and $\mathbf b$ is $45^\circ$, find the exact values of the following:

(i) $\mathbf a\cdot\mathbf b$

(ii) $(4\mathbf a+2\mathbf b)\cdot(3\mathbf a-\mathbf b)$

(iii) $|\mathbf a-\mathbf b|^2$

The angle between two unit vectors $\mathbf a$ and $\mathbf b$ is $\theta$.

By expanding the scalar product, show that $(3\mathbf a-\mathbf b)\cdot(\mathbf a+3\mathbf b)=8\cos\theta$.

Given that $\theta=60^\circ$, show that the length of projection of $(3\mathbf a-\mathbf b)$ onto $(\mathbf a+3\mathbf b)$ is $\dfrac4{\sqrt{13}}$.

$A$, $B$ 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 $|\mathbf a\times\mathbf b|$.