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)  \(\theta\approx54.14^\circ\approx54^\circ8'\)
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)  \(\theta\approx69.80^\circ\approx69^\circ48'\)
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)  \(\theta\approx144.74^\circ\approx144^\circ44'\)
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:    \(\alpha\approx58^\circ2'\),   \(\beta\approx61^\circ56'\),   \(\gamma\approx45^\circ6'\)
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:    \(\alpha\approx43^\circ1'\)
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:    \(\alpha\approx40^\circ54'\),   \(\beta=60^\circ\),   \(\gamma\approx79^\circ6'\)
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:    \(\alpha=\gamma\approx35^\circ44'\),   \(\beta=\delta\approx144^\circ16'\)
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:    \(\varphi\approx74^\circ57'\)
20. Find the acute angle between the diagonals \(AG\) and \(BH\) in the cube \(ABCDEFGH\). Round the result to three significant figures.
    Solutions:    \(\varphi\approx70.5^\circ\)
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:    \(\varphi\approx56.5^\circ\)
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)  \(\theta\approx76^\circ16'\)
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)  \(d(A,l)=3\sqrt{5}\approx6.71\)
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 \(t=3.75~\mathrm{h}=3~\mathrm{h}~45~\mathrm{min}\)
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)  \(d\approx1704~\mathrm{km}\);     (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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