Chapter Questions
With reference to three axes drawn mutually at right angles, depict the vectors(a) $o p=4 i+3 j-2 k$ and (b) or $=5 i-2 j+2 k$
Find vector $a$ joining points $P$ and $Q$ where point $P$ has co-ordinates $(4,-1,3)$ and point $Q$ has co-ordinates $(2,5,0)$. Also, find $|a|$, the magnitude or norm of $a$
If $\boldsymbol{p}=2 \boldsymbol{i}+\boldsymbol{j}-\boldsymbol{k}$ and $\boldsymbol{q}=\boldsymbol{i}-3 \boldsymbol{j}+\boldsymbol{k}$ determine:(a) $p \cdot q$(b) $p+q$(c) $|p+q|$(d) $|p|+|q|$
Determine the angle between vectors oa and $o b$ when$$\begin{aligned}o a &=i+2 j-3 k \\\text { and } & o b=2 i-j+4 k\end{aligned}$$
Find the direction cosines of 3$$i+2 \boldsymbol{j}+\boldsymbol{k}$$
A constant force of $\boldsymbol{F}=10 \boldsymbol{i}+2 \boldsymbol{j}-\boldsymbol{k}$ newtons displaces an object from $\boldsymbol{A}=\boldsymbol{i}+\boldsymbol{j}+\boldsymbol{k}$ to $\boldsymbol{B}=2 \boldsymbol{i}-\boldsymbol{j}+3 \boldsymbol{k}$ (in metres). Find the work done in newton metres.
For the vectors $\boldsymbol{a}=\boldsymbol{i}+4 \boldsymbol{j}-2 \boldsymbol{k}$ and $b=2 i-j+3 k$ find (a) $a \times b$ and $(b)|a \times b|$
. If $\boldsymbol{p}=4 \boldsymbol{i}+\boldsymbol{j}-2 \boldsymbol{k}, \boldsymbol{q}=3 \boldsymbol{i}-2 \boldsymbol{j}+\boldsymbol{k}$ and $\boldsymbol{r}=\boldsymbol{i}-2 \boldsymbol{k}$ find $(\mathrm{a})(\boldsymbol{p}-2 \boldsymbol{q}) \times \boldsymbol{r} \quad$ (b) $p \times(2 \boldsymbol{r} \times 3 \boldsymbol{q})$
Find the moment and the magnitude of the moment of a force of $(\boldsymbol{i}+2 \boldsymbol{j}-3 \boldsymbol{k})$ newtons about point $B$ having co-ordinates $(0,1,1)$, when the force acts on a line through $A$ whose co-ordinates are $(1,3,4)$
The axis of a circular cylinder coincides with the $z$-axis and it rotates with an angular velocity of $(2 i-5 j+7 k) \mathrm{rad} / \mathrm{s}$. Determine the tangential velocity at a point $P$ on the cylinder, whose co-ordinates are $(\boldsymbol{j}+3 \boldsymbol{k})$ metres, and also determine the magnitude of the tangential velocity.
(a) Determine the vector equation of the line through the point with position vector $2 \boldsymbol{i}+3 \boldsymbol{j}-\boldsymbol{k}$ which is parallel to the vector $\boldsymbol{i}-2 \boldsymbol{j}+3 \boldsymbol{k}$.(b) Find the point on the line corresponding to $\lambda=3$ in the resulting equation of part (a).(c) Express the vector equation of the line in standard Cartesian form.
The equation$$\frac{2 x-1}{3}=\frac{y+4}{3}=\frac{-z+5}{2}$$represents a straight line. Express this in vector form.