Review exercises  —  vectors

1. Consider vectors $\mathbf{a}=\left(\begin{array}{c}4\\ 3\end{array}\right)$,  $\mathbf{b}=\left(\begin{array}{c}-2\\ 2\end{array}\right)$  and  $\mathbf{c}=\left(\begin{array}{c}2\\ 19\end{array}\right)$.
   

   (a)   Express vector $\mathbf{c}$ in the form $\mathbf{c}=n\mathbf{a}+m\mathbf{b}$.

   (b)   Calculate $|\mathbf{a}|$ and $|\mathbf{b}|$.

   (c)   Calculate scalar product $\mathbf{a}\cdot \mathbf{b}$.

   (d)   Find the angle between $\mathbf{a}$ and $\mathbf{b}$.

   Solutions:    (a)  $\mathbf{c}=3\mathbf{a}+5\mathbf{b}$;     (b)  $|\mathbf{a}|=5$, $|\mathbf{b}|=2\sqrt{2}$;     (c)  $\mathbf{a}\cdot \mathbf{b}=-2$;     (d)  $\theta \approx {98.1}^{\circ }$
2. Points $A(1,3)$, $B(13,8)$ and $C(22,23)$ are three of the vertices of a paralellogram $ABCD$.
   

   (a)   Find the coordinates of the vertex $D$.

   (b)   Calculate angle $\alpha =B\hat{A}D$.

   (c)   Calculate the length of the longer diagonal.

   Solutions:    (a)  $D(10,18)$;     (b)  $\alpha \approx {36.4}^{\circ }$;     (c)  $AC=29$
3. Quadrilateral $ABCD$ has vertices $A(-3,1),B(9,4),C(7,9)$ and $D(-1,7)$.
   

   (a)   Show that this quadrilateral is not a parallelogram.

   (b)   Show that sides $AB$ and $CD$ are parallel.

   Straight line $L$ passes through points $A$ and $B$.

   (c)   Write the equation of $L$ in the form $y=mx+c$.

   Solutions:    (a)  $\overrightarrow{AB}\ne \overrightarrow{DC}$, $\overrightarrow{AD}\ne \overrightarrow{BC}$ ;     (b)  $\overrightarrow{AB}=\frac{3}{2}\overrightarrow{DC}$;     (c)  $y=\frac{1}{4}x+\frac{7}{4}$
4. Triangle $ABC$ has vertices $A(1,3,6),B(13,9,10)$ and $C(7,12,8)$.
   

   (a)   Calculate lengths of all three sides.

   (b)   Calculate the largest angle in this triangle.

   (c)   Calculate the area of this triangle.

   Solutions:    (a)  $a=BC=7,b=AC=11,c=AB=14$ ;     (b)  $\gamma =A\hat{C}B\approx {99.7}^{\circ }$;     (c)  $A\approx 37.9$
5. Triangle $ABC$ has $A(-1,3)$ and $\overrightarrow{AB}=\left(\begin{array}{c}9\\ -2\end{array}\right)$, $\overrightarrow{BC}=\left(\begin{array}{c}3\\ 6\end{array}\right)$.
   

   (a)   Find the coordinates of points $B$ and $C$.

   (b)   Write down $\overrightarrow{BA}$.

   (c)   Calculate the angle $\beta =A\hat{B}C$.

   Point $D$ is the midpoint of the side $AC$.

   (d)   Calculate the distance $DB$.

   Solutions:    (a)  $B(8,1),C(11,7)$;     (b)  $\overrightarrow{BA}=\left(\begin{array}{c}-9\\ 2\end{array}\right)$;     (c)  $\beta \approx {104}^{\circ }$;     (d)  $DB=5$
6. Let  $\overrightarrow{PQ}=4\mathbf{i}+2\mathbf{j}+4\mathbf{k}$  and  $\overrightarrow{PR}=6\mathbf{i}+8\mathbf{j}-\mathbf{k}$.
   

   (a)   Write $\overrightarrow{QR}$.

   (b)   Show that $\overrightarrow{PQ}$ is perpendicular to $\overrightarrow{QR}$.

   Unit vector $\mathbf{u}$ has the same direction as $\overrightarrow{PQ}$.

   (c)   Write $\mathbf{u}$ in terms of $\mathbf{i},\mathbf{j}$ and $\mathbf{k}$.

   Solutions:    (a)  $\overrightarrow{QR}=\left(\begin{array}{c}2\\ 6\\ -5\end{array}\right)=2\mathbf{i}+6\mathbf{j}-5\mathbf{k}$;     (b)  $\overrightarrow{PQ}\cdot \overrightarrow{QR}=0$;     (c)  $\mathbf{u}=\frac{2}{3}\mathbf{i}+\frac{1}{3}\mathbf{j}+\frac{2}{3}\mathbf{k}$
7. Vectors  $\mathbf{a}=\left(\begin{array}{c}5\\ 14\\ 2\end{array}\right)$  and  $\mathbf{b}=\left(\begin{array}{c}10\\ 10\\ p\end{array}\right)$ have equal lengths.
   

   (a)   Find $p$, given that $p$ is a positive number.

   Consider two other vectors:  $\mathbf{c}=\mathbf{a}+\mathbf{b}$  and  $\mathbf{d}=\mathbf{a}-\mathbf{b}$.

   (b)   Write down $\mathbf{c}$ and $\mathbf{d}$.

   (c)   Show that $\mathbf{c}$ is perpendicular to $\mathbf{d}$.

   Solutions:    (a)  $p=5$;     (b)  $\mathbf{c}=\left(\begin{array}{c}15\\ 24\\ 7\end{array}\right)$,  $\mathbf{d}=\left(\begin{array}{c}-5\\ 4\\ -3\end{array}\right)$;     (c)  $\mathbf{c}\cdot \mathbf{d}=0$
8. Line $L$ passes through points $P(5,1)$ and $Q(9,5)$.
   

   (a)   Write down $\overrightarrow{PQ}$.

   (b)   Write the equation of $L$ in the form $\mathbf{r}=\mathbf{a}+\mathbf{b}t$.

   (c)   Calculate the acute angle formed by $L$ and the horizontal axis.

   Solutions:    (a)  $\overrightarrow{PQ}=\left(\begin{array}{c}4\\ 4\end{array}\right)$;     (b)  $\mathbf{r}=\left(\begin{array}{c}5\\ 1\end{array}\right)+\left(\begin{array}{c}4\\ 4\end{array}\right)t$;     (c)  $\phi ={45}^{\circ }$
9. Line $L_1$ passes through points $A(4,5,-1)$ and $B(6,8,0)$.
   

   (a)   Find $\overrightarrow{AB}$.

   (b)   Write the equation of $L_1$ in the form $\mathbf{r}=\mathbf{a}+\mathbf{b}t$.

   Line $L_2$ has the equation  $\mathbf{r}=\left(\begin{array}{c}12\\ 10\\ 0\end{array}\right)+\left(\begin{array}{c}1\\ 5\\ 2\end{array}\right)t$.

   (c)   Find the angle between $L_1$ and $L_2$.

   (d)   Find the intersection point.

   Solutions:    (a)  $\overrightarrow{AB}=\left(\begin{array}{c}2\\ 3\\ 1\end{array}\right)$;     (b)  $\mathbf{r}=\left(\begin{array}{c}4\\ 5\\ -1\end{array}\right)+\left(\begin{array}{c}2\\ 3\\ 1\end{array}\right)t$;     (c)  $\phi \approx {22.0}^{\circ }$;     (d)  $P(14,20,4)$
10. Line $L$ passes through points $A(4,2)$ and is perpendicular to vector $\mathbf{v}=\left(\begin{array}{c}3\\ 4\end{array}\right)$.
    

    (a)   Write the equation of $L$ in the form $\mathbf{r}=\mathbf{a}+\mathbf{b}t$.

    (b)   Write the equation of $L$ in the form $y=mx+c$.

    (c)   Write the equation of $L$ in the form $px+qy=s$, where $p,q$ and $s$ are integers.

    Solutions:    (a)  $\mathbf{r}=\left(\begin{array}{c}4\\ 2\end{array}\right)+\left(\begin{array}{c}4\\ -3\end{array}\right)t$;     (b)  $y=-\frac{3}{4}x+5$;     (c)  $3x+4y=20$
11. Line $L_1$ has the equation  $\mathbf{r}=(2+3t)\mathbf{i}+(-1+t)\mathbf{j}$.
    

    (a)   Write the equation of $L_1$ in the form $\mathbf{r}=\mathbf{a}+\mathbf{b}t$.

    (b)   Show that $L_1$ passes through the point $A(11,2)$.

    Line $L_2$ passes through $A$ and is perpendicular to $L_1$.

    (c)   Write the equation of $L_2$ in the form $y=mx+c$.

    Solutions:    (a)  $\mathbf{r}=\left(\begin{array}{c}2\\ -1\end{array}\right)+\left(\begin{array}{c}3\\ 1\end{array}\right)t$;     (c)  $y=-3x+35$

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