1If θ is the angle between vectors \( \vec{a} \) and \( \vec{b} \), which expression is their scalar product: \( |\vec{a}||\vec{b}|\sin\theta \), \( |\vec{a}||\vec{b}|\cos\theta \) or \( |\vec{a}| + |\vec{b}|\cos\theta \)?
2If \( \vec{a} = (x_1, y_1) \) and \( \vec{b} = (x_2, y_2) \), what is \( \vec{a}\cdot\vec{b} \)?
3If \( \vec{a}\cdot\vec{b} = 0 \), \( |\vec{a}| \neq 0 \) and \( |\vec{b}| \neq 0 \), what is the angle between \( \vec{a} \) and \( \vec{b} \)?
4If \( \vec{a} \) and \( \vec{b} \) are parallel and point in the same direction, what is \( \vec{a}\cdot\vec{b} \)?
5If \( \vec{a} = \begin{pmatrix} 2 \\ 3 \end{pmatrix} \) and \( \vec{b} = \begin{pmatrix} -3 \\ 2 \end{pmatrix} \), find \( \vec{a}\cdot\vec{b} \).
6If \( \vec{a} = \begin{pmatrix} 2 \\ 3 \end{pmatrix} \), what is the value of \( (\vec{a})^2 \)?
7If \( \overrightarrow{OA} = \begin{pmatrix} 5 \\ m \end{pmatrix} \), \( \overrightarrow{OB} = \begin{pmatrix} 4 \\ -2 \end{pmatrix} \) and \( \angle AOB = 90^\circ \), what is m?
8What is the scalar product of two vectors? Explain with an example.
9Under what conditions are two vectors perpendicular or parallel? Give reasons.
10Find the scalar product of \( \vec{i} = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \) and \( \vec{j} = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \).
11Find the scalar product of \( \vec{a} = \begin{pmatrix} 2 \\ 0 \end{pmatrix} \) and \( \vec{b} = \begin{pmatrix} 3 \\ 0 \end{pmatrix} \).
12Find the scalar product of \( \vec{a} = \begin{pmatrix} 4 \\ 2 \end{pmatrix} \) and \( \vec{b} = \begin{pmatrix} 1 \\ 2 \end{pmatrix} \).
13Find the scalar product of \( \vec{p} = \begin{pmatrix} 3 \\ 1 \end{pmatrix} \) and \( \vec{q} = \begin{pmatrix} 3 \\ -1 \end{pmatrix} \).
14Find the angle between \( \vec{i} = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \) and \( \vec{j} = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \).
15Find the angle between \( \vec{a} = \begin{pmatrix} 2 \\ 0 \end{pmatrix} \) and \( \vec{b} = \begin{pmatrix} 0 \\ 3 \end{pmatrix} \).
16Find the angle between \( \vec{a} = \begin{pmatrix} -4 \\ 2 \end{pmatrix} \) and \( \vec{b} = \begin{pmatrix} 1 \\ 2 \end{pmatrix} \).
17Find the angle between \( \vec{p} = \begin{pmatrix} 3 \\ 1 \end{pmatrix} \) and \( \vec{q} = \begin{pmatrix} 3 \\ -3 \end{pmatrix} \).
18Show whether \( \vec{i} = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \) and \( \vec{j} = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \) are perpendicular or parallel.
19Show whether \( \vec{a} = \begin{pmatrix} 2 \\ 0 \end{pmatrix} \) and \( \vec{b} = \begin{pmatrix} 0 \\ 3 \end{pmatrix} \) are perpendicular or parallel.
20If \( \vec{a} = \begin{pmatrix} 2 \\ m \end{pmatrix} \) and \( \vec{b} = \begin{pmatrix} 3 \\ 2 \end{pmatrix} \) are perpendicular, find m.
21If \( \vec{a} = \begin{pmatrix} 3 \\ m \end{pmatrix} \) and \( \vec{b} = \begin{pmatrix} 6 \\ 4 \end{pmatrix} \) are perpendicular, find m.
22If \( \vec{a} = 3\vec{i} + m\vec{j} \) and \( \vec{b} = 6\vec{i} - 2\vec{j} \) are perpendicular, find m.
23If \( 3\vec{i} - 2\vec{j} \) and \( m\vec{i} + 3\vec{j} \) are perpendicular, find m.
24If \( \vec{a} = \begin{pmatrix} 2 \\ m \end{pmatrix} \) and \( \vec{b} = \begin{pmatrix} 3 \\ 2 \end{pmatrix} \) are parallel, find m.
25If \( \begin{pmatrix} 3 \\ m \end{pmatrix} \) and \( \begin{pmatrix} 6 \\ 4 \end{pmatrix} \) are parallel, find m.
26If \( \vec{a} = 3\vec{i} + m\vec{j} \) and \( \vec{b} = 6\vec{i} - 2\vec{j} \) are parallel, find m.
27If \( 3\vec{i} - 2\vec{j} \) and \( m\vec{i} + 3\vec{j} \) are parallel, find m.
28For \( \vec{a} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix} \) and \( \vec{b} = \begin{pmatrix} x_2 \\ y_2 \end{pmatrix} \), what is the scalar product \( \vec{a}\cdot\vec{b} \)?
29If \( \vec{a}\cdot\vec{b} = 0 \), what is the relationship between \( \vec{a} \) and \( \vec{b} \)?
30If \( \vec{a} = \begin{pmatrix} 2 \\ 3 \end{pmatrix} \) and \( \vec{b} = \begin{pmatrix} 4 \\ 5 \end{pmatrix} \), find \( \vec{a}\cdot\vec{b} \).
31Under what condition is \( \vec{a}\cdot\vec{b} \) greatest? Explain with reason.
32Four points A(-2, 1), B(-1, -3), C(3, -2) and D(2, 2) are given. If \( A(x_1, y_1) \) and \( B(x_2, y_2) \), what is the vector \( \overrightarrow{AB} \)?
33For A(-2, 1), B(-1, -3), C(3, -2) and D(2, 2), write \( \overrightarrow{AB} \) and \( \overrightarrow{BD} \) as column vectors.
34For A(-2, 1), B(-1, -3), C(3, -2) and D(2, 2), find the angle between \( \overrightarrow{AC} \) and \( \overrightarrow{BD} \).
35Is ABCD, with A(-2, 1), B(-1, -3), C(3, -2) and D(2, 2), a rectangle or a square? Justify using the lengths of the sides and the angles.
36It is given that \( \vec{a} + \vec{b} + \vec{c} = 0 \). What does this mean?
37If \( \vec{a} + \vec{b} + \vec{c} = 0 \), \( |\vec{a}| = 3 \), \( |\vec{b}| = 5 \) and \( |\vec{c}| = 4 \), find \( \vec{a}\cdot\vec{c} \).
38If \( |\vec{a} + \vec{b}| = |\vec{a} - \vec{b}| \), prove that \( \vec{a} \) and \( \vec{b} \) are perpendicular.
39For the triangle with P(2, 1), Q(6, 1) and R(6, 4), write \( \overrightarrow{PQ} \), \( \overrightarrow{QR} \) and \( \overrightarrow{PR} \) in components and find the relationship between them.
40How can \( \overrightarrow{PQ} \) and \( \overrightarrow{QR} \) each be expressed in terms of two other vectors?
41If \( |\vec{a}| = 4 \) and \( \vec{a}\cdot\vec{b} = 10 \), find the projection of \( \vec{b} \) on \( \vec{a} \).
42Show that \( \vec{a} = (4, 6) \) and \( \vec{b} = (6, 9) \) are parallel, and say whether they point the same way.
43Explain why \( \vec{i}\cdot\vec{i} = 1 \) but \( \vec{i}\cdot\vec{j} = 0 \).
44Without finding the angle, decide whether the angle between \( (2, -5) \) and \( (3, 1) \) is acute, right or obtuse.