Scalar product

1. Find the scalar product $\mathbf{a}\cdot \mathbf{b}$ given both lengths and the angle between the vectors:
   

   (a)   $|\mathbf{a}|=3,|\mathbf{b}|=4,\theta ={60}^{\circ }$

   (b)   $|\mathbf{a}|=5,|\mathbf{b}|=6,\theta ={120}^{\circ }$

   (c)   $|\mathbf{a}|=3,|\mathbf{b}|=5,\theta ={90}^{\circ }$

   Solutions:    (a)  $\mathbf{a}\cdot \mathbf{b}=6$;     (b)  $\mathbf{a}\cdot \mathbf{b}=-15$;     (c)  $\mathbf{a}\cdot \mathbf{b}=0$
2. Find the scalar product of the following vectors:
   

   (a)   $\mathbf{a}=3\mathbf{i}+4\mathbf{j}$   and   $\mathbf{b}=\mathbf{i}+5\mathbf{j}$

   (b)   $\mathbf{e}=7\mathbf{i}-\mathbf{j}+\mathbf{k}$   and   $\mathbf{f}=\mathbf{i}+\mathbf{j}+2\mathbf{k}$

   (c)   $\mathbf{u}=2\mathbf{i}-4\mathbf{k}$   and   $\mathbf{v}=\mathbf{j}+6\mathbf{k}$

   Solutions:    (a)  $\mathbf{a}\mathbf{b}=23$;     (b)  $\mathbf{e}\mathbf{f}=8$;     (c)  $\mathbf{u}\mathbf{v}=-24$
3. Find the scalar product of the following vectors:
   

   (a)   $\mathbf{c}=\left(\begin{array}{c}2\\ 3\\ -1\end{array}\right)$   and   $\mathbf{d}=\left(\begin{array}{c}3\\ 1\\ 4\end{array}\right)$

   (b)   $\mathbf{w}=\left(\begin{array}{c}4\\ -6\\ 2\end{array}\right)$   and   $\mathbf{z}=\left(\begin{array}{c}5\\ 1\\ -7\end{array}\right)$

   Solutions:    (a)  $\mathbf{c}\mathbf{d}=5$;     (b)  $\mathbf{w}\mathbf{z}=0$
4. Let $\mathbf{a}=\left(\begin{array}{c}4\\ 3\\ 0\end{array}\right)$,   $\mathbf{b}=\left(\begin{array}{c}5\\ 2\\ -1\end{array}\right)$,   $\mathbf{c}=\left(\begin{array}{c}-2\\ 4\\ -3\end{array}\right)$.
   

   Evaluate the following scalar products:

   (a)   $\mathbf{a}\mathbf{c}$

   (b)   $\mathbf{a}\mathbf{a}$

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

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

   Solutions:    (a)  $\mathbf{a}\mathbf{c}=4$;     (b)  $\mathbf{a}\mathbf{a}=25$;     (c)  $\mathbf{a}(\mathbf{b}+\mathbf{c})=30$;     (d)  $(2\mathbf{a}+3\mathbf{b})(\mathbf{b}-2\mathbf{c})=120$
5. Consider vectors $\mathbf{a}=\left(\begin{array}{c}6\\ -9\\ 3\end{array}\right)$,   $\mathbf{b}=\left(\begin{array}{c}5\\ 2\\ -1\end{array}\right)$   and  $\mathbf{c}=\left(\begin{array}{c}-4\\ -1\\ 5\end{array}\right)$.
   

   Determine which two of them are perpendicular.

   Solutions:    Vectors $\mathbf{a}$ and $\mathbf{c}$ are perpendicular $(\mathbf{a}\mathbf{c}=0)$.
6. Vectors $\mathbf{u}=\mathbf{i}+2\mathbf{j}-\mathbf{k}$ and $\mathbf{v}=2\mathbf{i}+a\mathbf{j}+4\mathbf{k}$ are perpendicular. Find the value of the constant $a$.
   Solutions:    $a=1$
7. Let $\mathbf{u}=\left(\begin{array}{c}a+1\\ a\\ -2\end{array}\right)$   and  $\mathbf{v}=\left(\begin{array}{c}a-1\\ -2\\ a+2\end{array}\right)$.
   

   Find the value of the constant $a$ given that these two vectors are perpendicular. Write down all possible solutions.

   Solutions:    $a_1=5,a_2=-1$
8. Let $\mathbf{a}=\left(\begin{array}{c}2\\ 4\\ 3\end{array}\right)$   and  $\mathbf{b}=\left(\begin{array}{c}6\\ 12\\ m\end{array}\right)$.
   

   (a)   Find the value of $m$ given that $\mathbf{a}$ and $\mathbf{b}$ are perpendicular.

   (b)   Find the value of $m$ given that $\mathbf{a}$ and $\mathbf{b}$ are parallel.

   Solutions:    (a)  $m=-20$;     (b)  $m=9$
9. Let $\mathbf{a}=\left(\begin{array}{c}1\\ -4\\ 8\end{array}\right)$   and  $\mathbf{b}=\left(\begin{array}{c}-4\\ m\\ 7\end{array}\right)$.
   

   (a)   Find the value of $m$ given that $\mathbf{a}$ and $\mathbf{b}$ are perpendicular.

   (b)   Find the value of $m$ given that $|\mathbf{a}|=|\mathbf{b}|$.

   Solutions:    (a)  $m=13$;     (b)  $m_1=4,m_2=-4$
10. Let $\mathbf{a}=\left(\begin{array}{c}3\\ 5\end{array}\right)$.
    

    Find vector $\mathbf{b}$ perpendicular to vector $\mathbf{a}$ given that $|\mathbf{b}|=|\mathbf{a}|$. Write down all possible solutions.

    Solutions:    ${\mathbf{b}}_1=\left(\begin{array}{c}5\\ -3\end{array}\right),{\mathbf{b}}_2=\left(\begin{array}{c}-5\\ 3\end{array}\right)$
11. Let $\mathbf{a}=\left(\begin{array}{c}1\\ 8\\ 4\end{array}\right)$  and  $\mathbf{b}=\left(\begin{array}{c}6\\ 2\\ 9\end{array}\right)$.
    

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

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

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

    Solutions:    (a)  $\mathbf{a}\mathbf{b}=58$;     (b)  $|\mathbf{a}|=9,|\mathbf{b}|=11$;     (c)  $\theta \approx {54.14}^{\circ }\approx {54}^{\circ }{8}^{\prime }$
12. Find the angles between the following pairs of vectors:
    

    (a)   $\mathbf{a}=\left(\begin{array}{c}3\\ 4\\ 5\end{array}\right)$  and  $\mathbf{b}=\left(\begin{array}{c}7\\ 1\\ 0\end{array}\right)$

    (b)   $\mathbf{a}=\left(\begin{array}{c}6\\ 3\\ 2\end{array}\right)$  and  $\mathbf{b}=\left(\begin{array}{c}9\\ 1\\ -4\end{array}\right)$

    (c)   $\mathbf{a}=\left(\begin{array}{c}5\\ 3\\ -1\end{array}\right)$  and  $\mathbf{b}=\left(\begin{array}{c}4\\ -2\\ 3\end{array}\right)$

    Solutions:    (a)  $\theta ={60}^{\circ }$;     (b)  $\theta ={45}^{\circ }$;     (c)  $\theta \approx {69.80}^{\circ }\approx {69}^{\circ }{48}^{\prime }$
13. Find the angles between the following pairs of vectors:
    

    (a)   $\mathbf{u}=\mathbf{i}+2\mathbf{j}+\mathbf{k}$  and  $\mathbf{v}=5\mathbf{i}+3\mathbf{j}+4\mathbf{k}$

    (b)   $\mathbf{u}=\mathbf{i}-\mathbf{j}+4\mathbf{k}$  and  $\mathbf{v}=8\mathbf{i}+7\mathbf{j}-7\mathbf{k}$

    (c)   $\mathbf{u}=-\mathbf{i}+4\mathbf{j}+3\mathbf{k}$  and  $\mathbf{v}=3\mathbf{i}-3\mathbf{j}+5\mathbf{k}$

    (d)   $\mathbf{u}=\mathbf{i}+\mathbf{j}-\mathbf{k}$  and  $\mathbf{v}=\mathbf{k}-\mathbf{i}$

    Solutions:    (a)  $\theta ={30}^{\circ }$;     (b)  $\theta ={120}^{\circ }$;     (c)  $\theta ={90}^{\circ }$;     (d)  $\theta \approx {144.74}^{\circ }\approx {144}^{\circ }{44}^{\prime }$
14. Let $\mathbf{a}=\left(\begin{array}{c}9\\ 8\\ 12\end{array}\right)$.
    

    The vector $\mathbf{a}$ forms the angle $\alpha$ with the $x$-axis, the angle $\beta$ with the $y$-axis and the angle $\gamma$ with the $z$-axis. Calculate the angles $\alpha ,\beta$ and $\gamma$. Round them to degrees and minutes.

    Hint:    Calculate the angles between $\mathbf{a}$ and the vectors $\mathbf{i},\mathbf{j}$ and $\mathbf{k}$.
    Solutions:    $\alpha \approx {58}^{\circ }{2}^{\prime }$,   $\beta \approx {61}^{\circ }{56}^{\prime }$,   $\gamma \approx {45}^{\circ }{6}^{\prime }$
15. Points $A(5,1),B(3,15)$ and $C(8,5)$ are the vertices of the triangle $ABC$. Calculate the angle between vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$.
    Solutions:    $\theta ={45}^{\circ }$
16. Points $A(3,1,5),B(10,3,8)$ and $C(7,9,11)$ are the vertices of the triangle $ABC$. Calculate the angle $\alpha =B\hat{A}C$.
    Solutions:    $\alpha \approx {43}^{\circ }{1}^{\prime }$
17. Points $A(2,6,7),B(12,11,6)$ and $C(6,15,8)$ are the vertices of the triangle $ABC$. Calculate all three angles of this triangle.
    Solutions:    $\alpha \approx {40}^{\circ }{54}^{\prime }$,   $\beta ={60}^{\circ }$,   $\gamma \approx {79}^{\circ }{6}^{\prime }$
18. Points $B(17,10,3),C(22,19,2)$ and $D(7,12,4)$ are three of the vertices of the parallelogram $ABCD$. Calculate all four angles of this parallelogram. Round the results to degrees and minutes.
    Solutions:    $\alpha =\gamma \approx {35}^{\circ }{44}^{\prime }$,   $\beta =\delta \approx {144}^{\circ }{16}^{\prime }$
19. Points $A(2,1,3),B(8,2,1)$ and $C(10,9,5)$ are three of the vertices of the parallelogram $ABCD$. Calculate the acute angle between the diagonals of this parallelogram. Round the result to degrees and minutes.
    Solutions:    $\phi \approx {74}^{\circ }{57}^{\prime }$
20. Find the acute angle between the diagonals $AG$ and $BH$ in the cube $ABCDEFGH$. Round the result to three significant figures.
    Solutions:    $\phi \approx {70.5}^{\circ }$
21. The cuboid $ABCDEFGH$ has the edges: $AB=7\mathrm{cm},BC=5\mathrm{cm}$ and $CG=13\mathrm{cm}$. Find the acute angle between the face diagonal $DB$ and the space diagonal $DF$. Round the result to three significant figures.
    Solutions:    $\phi \approx {56.5}^{\circ }$
22. Let $\mathbf{a}=\left(\begin{array}{c}m\\ -5\sqrt{3}\\ 12\end{array}\right)$.
    

    Determine the value of $m$ so that the angle between the vectors $\mathbf{a}$ and $\mathbf{j}$ will be ${120}^{\circ }$.

    Solutions:    $m_1=9,m_2=-9$
23. Let $\mathbf{a}=\left(\begin{array}{c}m\\ -1\\ 0\end{array}\right)$,  and  $\mathbf{b}=\left(\begin{array}{c}4\\ 3\\ 5\end{array}\right)$.
    

    Determine the value of $m$ so that the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$ will be ${60}^{\circ }$.

    Solutions:    $m=7$

Vector equation of a line

24. Points $P$ and $Q$ have the coordinates $P(1,5)$ and $Q(4,7)$.
    

    (a)   Write the vector equation of the line passing through $P$ and $Q$.

    (b)   Does the point $A(10,11)$ lie on this line?

    (c)   Does the point $B(7,10)$ lie on this line?

    Solutions:    (a)  $\mathbf{r}=\left(\begin{array}{c}1\\ 5\end{array}\right)+t\left(\begin{array}{c}3\\ 2\end{array}\right)$;     (b)  yes;     (c)  no
25. The line $l$ has the equation:  $\mathbf{r}=\left(\begin{array}{c}1\\ 5\end{array}\right)+t\left(\begin{array}{c}3\\ 2\end{array}\right)$.
    

    Find out which of the following points lie on the line $l$:

    (a)   $A(10,11)$

    (b)   $B(4,7)$

    (c)   $C(16,13)$

    (d)   $D(-5,1)$

    Solutions:    (a)  yes;     (b)  yes;     (c)  no;     (d)  yes
26. The line $l$ has the equation:  $\mathbf{r}=\left(\begin{array}{c}2\\ 5\\ 1\end{array}\right)+t\left(\begin{array}{c}4\\ 3\\ -1\end{array}\right)$.
    

    Find out which of the following points lie on the line $l$:

    (a)   $A(-2,2,2)$

    (b)   $B(10,10,-10)$

    (c)   $C(14,14,14)$

    Solutions:    (a)  yes;     (b)  no;     (c)  no
27. Consider points $A(8,6,10)$ and $B(12,9,15)$.
    

    (a)   Write the vector equation of the line joining $A$ and $B$.

    (b)   Does this line pass through the point $C(2,4,6)$?

    (c)   Does this line pass through the origin of the coordinate system?

    Solutions:    (a)  $\mathbf{r}=\left(\begin{array}{c}8\\ 6\\ 10\end{array}\right)+t\left(\begin{array}{c}4\\ 3\\ 5\end{array}\right)$;     (b)  no;     (c)  yes
28. Let $\mathbf{v}=\left(\begin{array}{c}4\\ -1\end{array}\right)$.
    

    Write down the vector equation of the line which passes through the point $P(3,2)$ and

    (a)   which is parallel to vector $\mathbf{v}$,

    (b)   which is perpendicular to vector $\mathbf{v}$.

    Solutions:    (a)  $\mathbf{r}=\left(\begin{array}{c}3\\ 2\end{array}\right)+t\left(\begin{array}{c}4\\ -1\end{array}\right)$;     (b)  $\mathbf{r}=\left(\begin{array}{c}3\\ 2\end{array}\right)+t\left(\begin{array}{c}1\\ 4\end{array}\right)$
29. Points $A(2,-3)$ and $B(10,7)$ are the endpoints of a line segment. Write the equation of the perpendicular bisector of this line segment.
    Solutions:    $\mathbf{r}=\left(\begin{array}{c}6\\ 2\end{array}\right)+t\left(\begin{array}{c}-10\\ 8\end{array}\right)$    or   $\mathbf{r}=\left(\begin{array}{c}6\\ 2\end{array}\right)+t\left(\begin{array}{c}-5\\ 4\end{array}\right)$
30. Three lines have the following equations:
    

    $l_1:\mathbf{r}=\left(\begin{array}{c}2\\ 7\end{array}\right)+t\left(\begin{array}{c}1\\ 2\end{array}\right)$,     $l_2:\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}3\\ 3\end{array}\right)+u\left(\begin{array}{c}6\\ 4\end{array}\right)$     $l_3:\mathbf{r}=(-1+s)\mathbf{i}+(4-3s)\mathbf{j}$

    Write the equations of these lines in form $y=mx+c$.

    Solutions:    $l_1:y=2x+3$,     $l_2:y=\frac{2}{3}x+1$,     $l_3:y=-3x+1$,    
31. Two lines $l_1$ and $l_2$ have the equations:
    

    $\mathbf{r}=\left(\begin{array}{c}4\\ -4\\ -9\end{array}\right)+t\left(\begin{array}{c}1\\ 2\\ 4\end{array}\right)$,     $\mathbf{r}=\left(\begin{array}{c}13\\ 12\\ 1\end{array}\right)+u\left(\begin{array}{c}3\\ 5\\ -1\end{array}\right)$

    Determine the point of intersection of these two lines if possible.

    Solutions:    $P(7,2,3)$
32. Two lines $l_1$ and $l_2$ have the equations:
    

    $\mathbf{r}=\left(\begin{array}{c}1\\ 5\\ 1\end{array}\right)+t\left(\begin{array}{c}4\\ 2\\ 1\end{array}\right)$,     $\mathbf{r}=\left(\begin{array}{c}11\\ 19\\ 13\end{array}\right)+u\left(\begin{array}{c}2\\ 4\\ 3\end{array}\right)$

    Determine the point of intersection of these two lines if possible.

    Solutions:    The intersection point doesn't exist – the lines are not concurrent.
33. Two lines have the equations:
    

    $l_1:\mathbf{r}=\left(\begin{array}{c}1\\ -1\\ 3\end{array}\right)+t\left(\begin{array}{c}6\\ 2\\ 4\end{array}\right)$,     $l_2:\mathbf{r}=\left(\begin{array}{c}2\\ 2\\ 7\end{array}\right)+t\left(\begin{array}{c}-1\\ 1\\ 1\end{array}\right)$

    (a)   Are these two lines concurrent?

    (b)   Are these two lines perpendicular?

    Solutions:    (a)  Yes, they are concurrent. They intersect at $P(4,0,5)$.     (b)  Yes, they are perpendicular.
34. Two lines have the equations:
    

    $l_1:\mathbf{r}=\left(\begin{array}{c}-1\\ 10\\ 3\end{array}\right)+t\left(\begin{array}{c}-3\\ 4\\ 2\end{array}\right)$,     $l_2:\mathbf{r}=\left(\begin{array}{c}-8\\ 4\\ -3\end{array}\right)+t\left(\begin{array}{c}5\\ 1\\ 2\end{array}\right)$

    (a)   Show that they are concurrent.

    (b)   Calculate the acute angle between these two lines. Round the result to nearest minute.

    Solutions:    (a)  They intersect at $P(2,6,1)$.     (b)  $\theta \approx {76}^{\circ }{16}^{\prime }$
35. Two lines have the equations:
    

    $l_1:\mathbf{r}=\left(\begin{array}{c}5\\ 3\\ 2\end{array}\right)+t\left(\begin{array}{c}1\\ 2\\ -1\end{array}\right)$,     $l_2:\mathbf{r}=\left(\begin{array}{c}6\\ 5\\ 1\end{array}\right)+t\left(\begin{array}{c}-3\\ -6\\ 3\end{array}\right)$

    Show that these two lines are coincident (both equations represent the same line).

    Solutions:    Their direction vectors are parallel and the point $P(5,3,2)$ lies on $l_2$ (or: $P(6,5,1)$ lies on $l_1$).
36. The point $A$ has the coordinates $A(12,8)$ and the line $l$ has the equation:
    

    $\mathbf{r}=\left(\begin{array}{c}3\\ 5\end{array}\right)+t\left(\begin{array}{c}1\\ 2\end{array}\right)$

    (a)   Find the point $B$ on $l$ so that $\overrightarrow{AB}$ is perpendicular to $l$.

    (b)   Find the perpendicular distance from $A$ to $l$.

    Solutions:    (a)  $B(6,11)$;     (b)  $d(A,l)=3\sqrt{5}\approx 6.71$
37. The point $A$ has the coordinates $A(6,7,-2)$ and the line $l$ has the equation:
    

    $\mathbf{r}=\left(\begin{array}{c}15\\ 9\\ 8\end{array}\right)+t\left(\begin{array}{c}4\\ 3\\ 1\end{array}\right)$

    (a)   Find the point $B$ on $l$ so that $\overrightarrow{AB}$ is perpendicular to $l$.

    (b)   Find the perpendicular distance from $A$ to $l$.

    Solutions:    (a)  $B(7,3,6)$;     (b)  $d(A,l)=9$
38. A boat starts its voyage at the origin $O(0,0)$. The position of the boat after $t$ hours is:
    

    $\left(\begin{array}{c}x\\ y\end{array}\right)=t\left(\begin{array}{c}16\\ 12\end{array}\right)$

    where the vector $\left(\begin{array}{c}0\\ 1\end{array}\right)$ represents the displacement of 1 km to the north.

    (a)   Find the position of this boat at $t=1$ and $t=2$.

    (b)   Find the speed of this boat.

    (c)   Calculate the time when the boat will reach the point $H(60,45)$.

    Solutions:    (a)  $P_1(16,12),P_2(32,24)$;     (b)  $|\mathbf{v}|=20$, so the speed is 20 km/h;     (c)  at $t=3.75\mathrm{h}=3\mathrm{h}45\min$
39. Two airplanes start their flights at the same time. Their positions after $t$ hours are given by the formulas:
    

    First airplane: $\mathbf{r}=\left(\begin{array}{c}380\\ 1340\end{array}\right)+t\left(\begin{array}{c}560\\ 330\end{array}\right)$

    Second airplane: $\mathbf{r}=\left(\begin{array}{c}2940\\ 350\end{array}\right)+t\left(\begin{array}{c}-480\\ 550\end{array}\right)$

    (a)   Find the speeds of both airplanes.

    (b)   Find the distance between them after 1 hour.

    (c)   Show that the courses of the two airplanes intersect. Write the coordinates of the point of intersection.

    (d)   Will these two airplanes collide?

    Solutions:    (a)  650 km/h and 730 km/h;     (b)  $d\approx 1704\mathrm{km}$;     (c)  $P(1500,2000)$;     (d)  no: the first plane will reach $P$ at $t=2$ and the second at $t=3$

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