Vectors in two dimensions

1. In the triangle \(\triangle ABC\) point \(P\) is the midpoint of the segment \(BC\). Write the following vectors in terms of vectors \(\mathbf{a}=\overrightarrow{AB}\) and \(\mathbf{b}=\overrightarrow{AC}\):
   

   (a)   \(\overrightarrow{CB}\)

   (b)   \(\overrightarrow{PC}\)

   (c)   \(\overrightarrow{AP}\)

   Solutions:    (a)  \(\overrightarrow{CB}=\mathbf{a}-\mathbf{b}\);     (b)  \(\overrightarrow{PC}=-\frac{1}{2}\mathbf{a}+\frac{1}{2}\mathbf{b}\);     (c)  \(\overrightarrow{AP}=\frac{1}{2}\mathbf{a}+\frac{1}{2}\mathbf{b}\)
2. The point \(S\) is the center of the regular hexagon \(ABCDEF\). Write the following vectors as linear combinations of vectors \(\mathbf{a}=\overrightarrow{AB}\) and \(\mathbf{b}=\overrightarrow{BC}\):
   

   (a)   \(\overrightarrow{AC}\)

   (b)   \(\overrightarrow{ES}\)

   (c)   \(\overrightarrow{FC}\)

   (d)   \(\overrightarrow{EA}\)

   Solutions:    (a)  \(\overrightarrow{AC}=\mathbf{a}+\mathbf{b}\);     (b)  \(\overrightarrow{ES}=\mathbf{a}-\mathbf{b}\);     (c)  \(\overrightarrow{FC}=2\mathbf{a}\);     (d)  \(\overrightarrow{EA}=\mathbf{a}-2\mathbf{b}\)
3. In the rectangle \(ABCD\) point \(P\) is the midpoint of the side \(CD\). Points \(U\) and \(V\) divide the side \(BC\) into three equal parts. Express the following vectors as linear combinations of given vectors \(\mathbf{a}=\overrightarrow{AB}\) and \(\mathbf{b}=\overrightarrow{AD}\):
   

   (a)   \(\overrightarrow{AC}\)

   (b)   \(\overrightarrow{BD}\)

   (c)   \(\overrightarrow{AP}\)

   (d)   \(\overrightarrow{AU}\)

   (e)   \(\overrightarrow{PV}\)

   Solutions:    (a)  \(\overrightarrow{AC}=\mathbf{a}+\mathbf{b}\);     (b)  \(\overrightarrow{BD}=-\mathbf{a}+\mathbf{b}\);     (c)  \(\overrightarrow{AP}=\frac{1}{2}\mathbf{a}+\mathbf{b}\);     (d)  \(\overrightarrow{AU}=\mathbf{a}+\frac{1}{3}\mathbf{b}\);     (e)  \(\overrightarrow{PV}=\frac{1}{2}\mathbf{a}-\frac{1}{3}\mathbf{b}\)
4. In the square \(ABCD\) point \(P\) is the midpoint of the side \(BC\). Express the following vectors as linear combinations of basis vectors \(\mathbf{a}=\overrightarrow{AC}\) and \(\mathbf{b}=\overrightarrow{AD}\):
   

   (a)   \(\overrightarrow{AB}\)

   (b)   \(\overrightarrow{DB}\)

   (c)   \(\overrightarrow{PD}\)

   Solutions:    (a)  \(\overrightarrow{AB}=\mathbf{a}-\mathbf{b}\);     (b)  \(\overrightarrow{DB}=\mathbf{a}-2\mathbf{b}\);     (c)  \(\overrightarrow{PD}=-\mathbf{a}+\frac{3}{2}\mathbf{b}\)
5. Write the vectors shown in the diagram bellow in column vector notation.
   

   

   Solutions:    \(\mathbf{a}=\left(\begin{array}{c} 2 \\ 2 \end{array}\right)\),     \(\mathbf{b}=\left(\begin{array}{c} 8 \\-1 \end{array}\right)\),     \(\mathbf{c}=\left(\begin{array}{c} 4 \\ 0 \end{array}\right)\),     \(\mathbf{d}=\left(\begin{array}{c} -6 \\ \frac{3}{2} \end{array}\right)\),     \(\mathbf{e}=\left(\begin{array}{c} -\frac{3}{2} \\ -4 \end{array}\right)\),     \(\mathbf{f}=\left(\begin{array}{c} 0 \\ -4 \end{array}\right)\)
6. Write the vectors shown in the diagram bellow in terms of standard basis vectors \(\mathbf{i}\) and \(\mathbf{j}\).
   

   

   Solutions:    \(\mathbf{a}=2\mathbf{i}+3\mathbf{j}\),     \(\mathbf{b}=6\mathbf{i}-3\mathbf{j}\),     \(\mathbf{c}=-\mathbf{i}+6\mathbf{j}\),     \(\mathbf{d}=-5\mathbf{i}\),     \(\mathbf{e}=6\mathbf{j}\),     \(\mathbf{f}=-7\mathbf{i}-3\mathbf{j}\)
7. Vectors \(\mathbf{a}\) and \(\mathbf{b}\) are shown in the diagram bellow.
   

   

   (a)   Draw vectors  \(\mathbf{a}+\mathbf{b}\)  and  \(\mathbf{a}-\mathbf{b}\).

   (b)   Write vectors \(\mathbf{a}\) and \(\mathbf{b}\) in column notation.

   (c)   Calculate vectors  \(\mathbf{a}+\mathbf{b}\)  and  \(\mathbf{a}-\mathbf{b}\) and write them in column notation.

   Solutions:    (b)  \(\mathbf{a}=\left(\begin{array}{c} 6 \\ 1 \end{array}\right)\),     \(\mathbf{b}=\left(\begin{array}{c} -2 \\ 3 \end{array}\right)\);     (c)  \(\mathbf{a}+\mathbf{b}=\left(\begin{array}{c} 4 \\ 4 \end{array}\right)\),     \(\mathbf{a}-\mathbf{b}=\left(\begin{array}{c} 8 \\ -2 \end{array}\right)\)
8. Vectors \(\mathbf{a}\) and \(\mathbf{b}\) have the coordinates:  \(\mathbf{a}=\left(\begin{array}{c} 10 \\ 6 \end{array}\right)\),    \(\mathbf{b}=\left(\begin{array}{c} 6 \\ 9 \end{array}\right)\).
   Calculate vectors:
   

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

   (b)   \(\mathbf{v}=5\mathbf{a}-6\mathbf{b}\)

   (c)   \(\mathbf{w}=-\frac{1}{2}\mathbf{a}+\frac{2}{3}\mathbf{b}\)

   Solutions:    (a)  \(\mathbf{u}=\left(\begin{array}{c} 16 \\ 15 \end{array}\right)\);     (b)  \(\mathbf{v}=\left(\begin{array}{c} 14 \\ -24 \end{array}\right)\);     (c)  \(\mathbf{w}=\left(\begin{array}{c} -1 \\ 3 \end{array}\right)\)
9. Given vectors \(\mathbf{a}=4\mathbf{i}-3\mathbf{j}\) and \(\mathbf{b}=5\mathbf{i}+\frac{3}{2}\mathbf{j}\) calculate:
   

   (a)   \(\mathbf{u}=\mathbf{a}+2\mathbf{b}\)

   (b)   \(\mathbf{v}=-\mathbf{a}+4\mathbf{b}\)

   (c)   \(\mathbf{w}=\frac{1}{2}\mathbf{a}-\mathbf{b}\)

   Solutions:    (a)  \(\mathbf{u}=14\mathbf{i}\);     (b)  \(\mathbf{v}=16\mathbf{i}+9\mathbf{j}\);     (c)  \(\mathbf{w}=-3\mathbf{i}-3\mathbf{j}\)
10. Given vector \(\mathbf{a}=\left(\begin{array}{c} 8 \\ 6 \end{array}\right)\) calculate:
    

    (a)   \(\frac{5}{2}\mathbf{a}\)

    (b)   \(2\mathbf{a}+3\mathbf{i}-\mathbf{j}\)

    (c)   \(|\mathbf{a}|\)

    Solutions:    (a)  \(\frac{5}{2}\mathbf{a}=\left(\begin{array}{c} 20 \\ 15 \end{array}\right)\);     (b)  \(2\mathbf{a}+3\mathbf{i}-\mathbf{j}=\left(\begin{array}{c} 19 \\ 11 \end{array}\right)\);     (c)  \(|\mathbf{a}|=10\)
11. Let \(\mathbf{a}=\left(\begin{array}{c} 21 \\ 72 \end{array}\right)\) and \(\mathbf{b}=\left(\begin{array}{c} 42 \\ -56 \end{array}\right)\).
    

    (a)   Find \(|\mathbf{a}|\) and \(|\mathbf{b}|\).

    (b)   Calculate \(\mathbf{c}=\mathbf{a}+\mathbf{b}\).

    (c)   Find \(|\mathbf{c}|\).

    Solutions:    (a)  \(|\mathbf{a}|=75,~~ |\mathbf{b}|=70\);     (b)  \(\mathbf{c}=\left(\begin{array}{c} 63 \\ 16 \end{array}\right)\);     (c)  \(|\mathbf{c}|=65\)
12. Let \(\mathbf{a}=\left(\begin{array}{c} 6 \\ 8 \end{array}\right)\).
    

    (a)   Find \(|\mathbf{a}|\).

    (b)   Write down the unit vector \(\mathbf{u}\) which has the direction of the vector \(\mathbf{a}\).

    Solutions:    (a)  \(|\mathbf{a}|=10\);     (b)  \(\mathbf{u}=\left(\begin{array}{c} \frac{3}{5} \\ \frac{4}{5} \end{array}\right)\)
13. Let \(\mathbf{a}=\left(\begin{array}{c} -5 \\ 12 \end{array}\right)\).
    Find the unit vector in the direction of the vector \(\mathbf{a}\) and write it in terms of \(\mathbf{i}\) and \(\mathbf{j}\).
    Solutions:    \(\mathbf{u}=-\frac{5}{13}\mathbf{i}+\frac{12}{13}\mathbf{j}\)
14. Consider vectors \(\mathbf{u}_1=\frac{1}{2}\mathbf{i}+\frac{\sqrt{3}}{2}\mathbf{j}\) and \(\mathbf{u}_2=\left(\begin{array}{c} -1 \\ 0 \end{array}\right)\).
    

    (a)   Show that \(\mathbf{u}_1\) and \(\mathbf{u}_2\) are unit vectors.

    (b)   Calculate \(|\mathbf{u}_1-\mathbf{u}_2|\) (write the exact value).

    Solutions:    (a)  \(|\mathbf{u}_1|=1,~~ |\mathbf{u}_2|=1\);     (b)  \(|\mathbf{u}_1-\mathbf{u}_2|=\sqrt{3}\)
15. Points \(A\) and \(B\) have coordinates \(A(3,1),~ B(5,6)\).
    

    (a)   Write down the position vectors \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\).

    (b)   Find the vector \(\overrightarrow{AB}\).

    Solutions:    (a)  \(\overrightarrow{OA}=\left(\begin{array}{c} 3 \\ 1 \end{array}\right)\),    \(\overrightarrow{OB}=\left(\begin{array}{c} 5 \\ 6 \end{array}\right)\);     (b)  \(\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}=\left(\begin{array}{c} 2 \\ 5 \end{array}\right)\)
16. Points \(A(-2,1),~ B(5,4)\) and \(C(3,8)\) are vertices of a triangle.
    

    (a)   Find vectors \(\mathbf{u}=\overrightarrow{AB}\),   \(\mathbf{v}=\overrightarrow{BC}\) and \(\mathbf{w}=\overrightarrow{CA}\).

    (b)   Verify that \(\mathbf{u}+\mathbf{v}+\mathbf{w}=\mathbf{0}\).

    Solutions:    (a)  \(\mathbf{u}=\left(\begin{array}{c} 7 \\ 3 \end{array}\right)\),    \(\mathbf{v}=\left(\begin{array}{c} -2 \\ 4 \end{array}\right)\),    \(\mathbf{w}=\left(\begin{array}{c} -5 \\ -7 \end{array}\right)\);     (b)  \(\mathbf{u}+\mathbf{v}+\mathbf{w}=\left(\begin{array}{c} 0 \\ 0 \end{array}\right)\)
17. Points \(A(2,3),~ B(9,5)\) and \(C(10,9)\) are three of the vertices of the parallelogram \(ABCD\).
    

    (a)   Find the coordinates of point \(D\).

    (b)   Find the length of the diagonal \(AC\)

    Solutions:    (a)  \(D(3,7)\);     (b)  \(|\overrightarrow{AC}|=10\)
18. Points \(A(-3,1),~ B(5,4)\) and \(D(-1,6)\) are three of the vertices of the parallelogram \(ABCD\).
    

    (a)   Find the coordinates of point \(C\).

    (b)   Find the coordinates of the intersection point of the diagonals \(AC\) and \(BD\).

    Solutions:    (a)  \(C(7,9)\);     (b)  \(P(2,5)\)
19. Points \(A(6,1)\) and \(B(16,5)\) are two of the vertices of the parallelogram \(ABCD\). Point \(M(13,8)\) is the midpoint of the side \(CD\). Point \(P\) is the intersection point of the diagonals.
    

    (a)   Find the coordinates of points \(C\) and \(D\).

    (b)   Find the coordinates of point \(P\).

    Solutions:    (a)  \(C(18,10),~ D(8,6)\);     (b)  \(P(12,5.5)\)
20. Points \(A(-5,0),~ B(22,-3),~ C(19,10)\) and \(D(-8,13)\) are the vertices of the quadrilateral \(ABCD\).
    

    (a)   Show that \(ABCD\) is a parallelogram.

    (b)   Find the coordinates of the point \(P\) which is the intersection point of the diagonals.

    (c)   Find the length of the longer diagonal.

    Solutions:    (a)  \(\overrightarrow{AB}=\overrightarrow{DC}=\left(\begin{array}{c} 27 \\ -3 \end{array}\right)\) and \(\overrightarrow{AD}=\overrightarrow{BC}=\left(\begin{array}{c} -3 \\ 13 \end{array}\right)\);     (b)  \(P(7,5)\);     (c)  \(|\overrightarrow{BD}|=34\)
21. A given line segment \(AB\) has the endpoints \(A(2,1)\) and \(B(8,10)\). Points \(U\) and \(V\) divide the segment \(AB\) in three equal parts. Find the coordinates of the points \(U\) and \(V\).
    Solutions:    \(U(4,4),~ V(6,7)\)
22. Line segment \(AB\) has the endpoints \(A(-3,2)\) and \(B(7,17)\). \(T\) is the point on \(AB\) such that \(AT:TB=2:3\). Find the coordinates of the point \(T\).
    Solutions:    \(T(1,8)\)
23. Points \(A(5,20)\) and \(B(41,-4)\) are the endpoints of the line segment \(AB\).
    

    (a)   Find the coordinates of the point \(U\) on \(AB\) such that \(AU:UB=1:3\).

    (b)   Find the coordinates of the point \(V\) on \(AB\) such that \(AV:AB=1:3\).

    Solutions:    (a)  \(U(14,14)\);     (b)  \(V(17,12)\)
24. \(M(22,31)\) is a point on the line segment \(AB\) such that \(AM:MB=2:5\). Given the point \(A(4,3)\) find the coordinates of the point \(B\).
    Solutions:    \(B(67,101)\)
25. Points \(B(17,5),~ C(25,21)\) and \(D(13,17)\) are three of the vertices of the parallelogram \(ABCD\). \(P\) is the point on \(AB\) such that \(AP:PB=1:3\), and \(R\) is the point on \(AD\) such that \(AR:RD=1:3\).
    

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

    (b)   Find the coordinates the points \(P\) and \(R\).

    (c)   Write down the coordinates the vector \(\overrightarrow{PR}\).

    (d)   Show that the vector \(\overrightarrow{PR}\) is parallel to the vector \(\overrightarrow{BD}\).

    Solutions:    (a)  \(A(5,1)\);     (b)  \(P(8,2),~ R(7,5)\);     (c)  \(\overrightarrow{PR}=\left(\begin{array}{c} -1 \\ 3 \end{array}\right)\);     (d)  \(\overrightarrow{BD}=4\cdot\overrightarrow{PR}\)
26. Consider vectors \(\mathbf{a}=\left(\begin{array}{c} 5 \\ 3 \end{array}\right)\),   \(\mathbf{b}=\left(\begin{array}{c} 1 \\ -2 \end{array}\right)\) and \(\mathbf{c}=\left(\begin{array}{c} 15 \\ -4 \end{array}\right)\). Express vector \(\mathbf{c}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).
    Solutions:    \(\mathbf{c}=2\mathbf{a}+5\mathbf{b}\)
27. Consider vectors \(\mathbf{u}=\left(\begin{array}{c} 8 \\ 3 \end{array}\right)\),   \(\mathbf{v}=\left(\begin{array}{c} -2 \\ 7 \end{array}\right)\) and \(\mathbf{w}=\left(\begin{array}{c} 13 \\ 1 \end{array}\right)\). Express vector \(\mathbf{w}\) in terms of \(\mathbf{u}\) and \(\mathbf{v}\).
    Solutions:    \(\mathbf{w}=\frac{3}{2}\mathbf{u}-\frac{1}{2}\mathbf{v}\)
28. Consider vectors \(\mathbf{a}=\left(\begin{array}{c} 2 \\ -3 \end{array}\right)\) and \(\mathbf{b}=\left(\begin{array}{c} -1 \\ 2 \end{array}\right)\). Write vector \(\mathbf{i}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).
    Solutions:    \(\mathbf{i}=2\mathbf{a}+3\mathbf{b}\)
29. Consider vectors \(\mathbf{u}=\left(\begin{array}{c} 6 \\ 9 \end{array}\right)\),   \(\mathbf{v}=\left(\begin{array}{c} 4 \\ 6 \end{array}\right)\) and \(\mathbf{w}=\left(\begin{array}{c} 8 \\ 10 \end{array}\right)\).
    

    (a)   Express vector \(\mathbf{u}\) in terms of \(\mathbf{v}\) and \(\mathbf{w}\) if possible.

    (b)   Express vector \(\mathbf{w}\) in terms of \(\mathbf{u}\) and \(\mathbf{v}\) if possible.

    Solutions:    (a)  \(\mathbf{u}=\frac{3}{2}\mathbf{v}+0\mathbf{w}\);     (b)  It's not possible.
30. Vector \(\mathbf{a}=\left(\begin{array}{c} 12 \\ m \end{array}\right)\) has the modulus \(|\mathbf{a}|=13\). Find the value of \(m\). Write down all possible solutions.
    Solutions:    \(m_1=5,~ m_2=-5\)

 Index