Vectors in three dimensions

1. The diagram shows a cube $ABCDEFGH$. Write the following vectors in terms of vectors $\mathbf{a}=\overrightarrow{AB},\mathbf{b}=\overrightarrow{BC}$ and $\mathbf{c}=\overrightarrow{CG}$:
   

   (a)   $\overrightarrow{AG}$

   (b)   $\overrightarrow{AH}$

   (c)   $\overrightarrow{CE}$

   (d)   $\overrightarrow{GD}$

   Solutions:    (a)  $\overrightarrow{AG}=\mathbf{a}+\mathbf{b}+\mathbf{c}$;     (b)  $\overrightarrow{AH}=\mathbf{b}+\mathbf{c}$;     (c)  $\overrightarrow{CE}=-\mathbf{a}-\mathbf{b}+\mathbf{c}$;     (d)  $\overrightarrow{GD}=-\mathbf{a}-\mathbf{c}$
2. The points $A(0,0,0),B(3,0,0),D(0,7,0)$ and $E(0,0,5)$ are vertices of the cuboid $ABCDEFGH$. Express the following vectors in terms of the standard basis vectors $\mathbf{i},\mathbf{j}$ and $\mathbf{k}$:
   

   (a)   $\overrightarrow{AC}$

   (b)   $\overrightarrow{AG}$

   (c)   $\overrightarrow{EC}$

   (d)   $\overrightarrow{HB}$

   Solutions:    (a)  $\overrightarrow{AC}=3\mathbf{i}+7\mathbf{j}$;     (b)  $\overrightarrow{AG}=3\mathbf{i}+7\mathbf{j}+5\mathbf{k}$;     (c)  $\overrightarrow{EC}=3\mathbf{i}+7\mathbf{j}-5\mathbf{k}$;     (d)  $\overrightarrow{HB}=3\mathbf{i}-7\mathbf{j}-5\mathbf{k}$
3. Vectors have the coordinates:  $\mathbf{a}=\left(\begin{array}{c}4\\ 1\\ 2\end{array}\right)$,    $\mathbf{b}=\left(\begin{array}{c}3\\ 5\\ 1\end{array}\right)$,    $\mathbf{c}=\left(\begin{array}{c}6\\ 2\\ -3\end{array}\right)$.
   Calculate vectors:
   

   (a)   $\mathbf{u}=\mathbf{a}+\mathbf{b}+\mathbf{c}$

   (b)   $\mathbf{v}=\mathbf{a}-\mathbf{b}-\mathbf{c}$

   (c)   $\mathbf{w}=2\mathbf{a}+3\mathbf{b}-\mathbf{c}$

   Solutions:    (a)  $\mathbf{u}=\left(\begin{array}{c}13\\ 8\\ 0\end{array}\right)$;     (b)  $\mathbf{v}=\left(\begin{array}{c}-5\\ -6\\ 4\end{array}\right)$;     (c)  $\mathbf{w}=\left(\begin{array}{c}11\\ 15\\ 10\end{array}\right)$
4. Let  $\mathbf{a}=8\mathbf{i}+7\mathbf{j}+4\mathbf{k}$  and  $\mathbf{b}=5\mathbf{i}+\mathbf{j}+6\mathbf{k}$.
   Calculate:
   

   (a)   $\mathbf{a}+\mathbf{b}$

   (b)   $\mathbf{a}-\mathbf{b}$

   (c)   $|\mathbf{a}-\mathbf{b}|$

   Solutions:    (a)  $\mathbf{a}+\mathbf{b}=13\mathbf{i}+8\mathbf{j}+10\mathbf{k}$;     (b)  $\mathbf{a}-\mathbf{b}=3\mathbf{i}+6\mathbf{j}-2\mathbf{k}$;     (c)  $|\mathbf{a}-\mathbf{b}|=7$
5. Let  $\mathbf{a}=\mathbf{i}-\mathbf{j}-\mathbf{k}$,    $\mathbf{b}=\mathbf{j}-\mathbf{i}-\mathbf{k}$  and  $\mathbf{c}=\mathbf{k}-\mathbf{j}$.
   Calculate:
   

   (a)   $\mathbf{a}+\mathbf{b}+\mathbf{c}$

   (b)   $\mathbf{b}-2\mathbf{a}-\mathbf{c}$

   Solutions:    (a)  $\mathbf{a}+\mathbf{b}+\mathbf{c}=-\mathbf{j}-\mathbf{k}$;     (b)  $\mathbf{b}-2\mathbf{a}-\mathbf{c}=-3\mathbf{i}+4\mathbf{j}$
6. Given vector  $\mathbf{a}=\left(\begin{array}{c}12\\ 8\\ 9\end{array}\right)$
   

   (a)   find $|\mathbf{a}|$,

   (b)   find the unit vector $\mathbf{u}$ which has the same direction as $\mathbf{a}$.

   Solutions:    (a)  $|\mathbf{a}|=17$;     (b)  $\mathbf{u}=\left(\begin{array}{c}\frac{12}{17}\\ \frac{8}{17}\\ \frac{9}{17}\end{array}\right)$
7. Let  $\mathbf{a}=4\mathbf{i}+\mathbf{j}-8\mathbf{k}$. Find the unit vector in the direction of the vector $\mathbf{a}$.
   Solutions:    $\mathbf{u}=\frac{4}{9}\mathbf{i}+\frac{1}{9}\mathbf{j}-\frac{8}{9}\mathbf{k}$
8. Points $P$ and $Q$ have the position vectors $\overrightarrow{OP}=5\mathbf{i}-\mathbf{j}+2\mathbf{k}$ and $\overrightarrow{OQ}=7\mathbf{i}+5\mathbf{j}-\mathbf{k}$.
   

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

   (b)   Find $|\overrightarrow{PQ}|$.

   Solutions:    (a)  $\overrightarrow{PQ}=2\mathbf{i}+6\mathbf{j}-3\mathbf{k}$;     (b)  $|\overrightarrow{PQ}|=7$
9. Points $B(5,2,3),C(7,6,2)$ and $D(1,8,1)$ are three of the vertices of the parallelogram $ABCD$.
   

   (a)   Find the coordinates of the point $A$.

   (b)   Find the coordinates of the intersection point of the diagonals.

   Solutions:    (a)  $A(-1,4,2)$;     (b)  $P(3,5,2)$
10. Points $A(-2,1,1)$ and $D(1,3,2)$ are two of the vertices of the parallelogram $ABCD$. Diagonals of this parallelogram intersect at $P(7,-1,3)$. Find the coordinates of points $B$ and $C$.
    Solutions:    $B(13,-5,4),C(16,-3,5)$
11. Let $\mathbf{v}=\left(\begin{array}{c}6\\ 4\\ 8\end{array}\right)$.   Find out which of the following vectors are parallel to $\mathbf{v}$:
    

    $\mathbf{a}=\left(\begin{array}{c}-12\\ -8\\ -16\end{array}\right)$,    $\mathbf{b}=\left(\begin{array}{c}24\\ 16\\ 30\end{array}\right)$,    $\mathbf{c}=\left(\begin{array}{c}9\\ 6\\ 12\end{array}\right)$

    Solutions:    $\mathbf{a}$ is parallel to $\mathbf{v}$,     $\mathbf{b}$ is not parallel to $\mathbf{v}$,     $\mathbf{c}$ is parallel to $\mathbf{v}$
12. Let $\mathbf{v}=\left(\begin{array}{c}5\\ -7\\ 5\end{array}\right)$,    $\mathbf{a}=\left(\begin{array}{c}1\\ 0\\ 2\end{array}\right)$,    $\mathbf{b}=\left(\begin{array}{c}2\\ 1\\ 0\end{array}\right)$,    $\mathbf{c}=\left(\begin{array}{c}0\\ 3\\ -1\end{array}\right)$.
    

    Express vector $\mathbf{v}$ in terms of $\mathbf{a},\mathbf{b}$ and $\mathbf{c}$.

    Solutions:    $\mathbf{v}=\mathbf{a}+2\mathbf{b}-3\mathbf{c}$
13. Let $\mathbf{a}=\left(\begin{array}{c}1\\ 2\\ 5\end{array}\right)$,    $\mathbf{b}=\left(\begin{array}{c}7\\ -1\\ 3\end{array}\right)$,    $\mathbf{c}=\left(\begin{array}{c}2\\ 6\\ 0\end{array}\right)$,    $\mathbf{d}=\left(\begin{array}{c}-4\\ 8\\ 7\end{array}\right)$.
    

    Express vector $\mathbf{d}$ in terms of $\mathbf{a},\mathbf{b}$ and $\mathbf{c}$.

    Solutions:    $\mathbf{d}=2\mathbf{a}-\mathbf{b}+\frac{1}{2}\mathbf{c}$
14. Let $\mathbf{a}=\left(\begin{array}{c}1\\ 1\\ 3\end{array}\right)$,    $\mathbf{b}=\left(\begin{array}{c}0\\ 2\\ 2\end{array}\right)$,    $\mathbf{c}=\left(\begin{array}{c}3\\ 7\\ 13\end{array}\right)$.
    

    Express vector $\mathbf{c}$ in terms of $\mathbf{a}$ and $\mathbf{b}$ if possible.

    Solutions:    It is possible: $\mathbf{c}=3\mathbf{a}+2\mathbf{b}$
15. Let $\mathbf{a}=\left(\begin{array}{c}2\\ 1\\ -3\end{array}\right)$,    $\mathbf{b}=\left(\begin{array}{c}1\\ 3\\ -1\end{array}\right)$,    $\mathbf{c}=\left(\begin{array}{c}3\\ -1\\ 2\end{array}\right)$.
    

    Express vector $\mathbf{c}$ in terms of $\mathbf{a}$ and $\mathbf{b}$ if possible.

    Solutions:    It is not possible.
16. Line segment $AB$ has the endpoints $A(-1,3,2)$ and $B(14,12,-1)$. Find the coordinates of the points $P$ and $Q$ which divide $AB$ in three equal parts.
    Solutions:    $P(4,6,1),Q(9,9,0)$
17. Points $U(3,5,-2)$ and $V(8,-5,8)$ are the endpoints of the line segment $UV$. Points $E,F,G$ and $H$ divide $UV$ in five equal parts. Find the coordinates of the points $E,F,G$ and $H$.
    Solutions:    $E(4,3,0),F(5,1,2),G(6,-1,4),H(7,-3,6)$
18. Points $A(3,2,-1)$ and $B(17,-5,6)$ are the endpoints of the line segment $AB$ and $P$ is the point on $AB$ such that $AP:PB=3:4$. Find the coordinates of the point $P$.
    Solutions:    $P(9,-1,2)$
19. Vector $\mathbf{a}=\left(\begin{array}{c}9\\ 2\\ z\end{array}\right)$ has the modulus $|\mathbf{a}|=11$. Find the unknown coordinate $z$. Write down all possible solutions.
    Solutions:    $z_1=6,z_2=-6$
20. Vector $\mathbf{v}=\mathbf{i}+m\mathbf{j}+2m\mathbf{k}$ has the modulus $|\mathbf{v}|=9$. Find the value of $m$. Write down all possible solutions.
    Solutions:    $m_1=4,m_2=-4$

 Index