Angles, perpendicular and parallel vectors Problem set B

Find the angle between each pair of vectors, to 3 significant figures.

1. (a) \(\mathbf{a} = \begin{pmatrix}3\\-2\\4\end{pmatrix},\ \mathbf{b} = \begin{pmatrix}-1\\5\\2\end{pmatrix}\)
2. (b) \(\mathbf{a} = 2\mathbf{i} + 6\mathbf{j} - 3\mathbf{k},\ \mathbf{b} = -4\mathbf{i} + \mathbf{j} + 5\mathbf{k}\)
3. (c) \(\mathbf{a} = 6\mathbf{i} - \mathbf{j},\ \mathbf{b} = 2\mathbf{i} + 7\mathbf{j}\)

The points P, Q and R have coordinates P(−3, 0, 2), Q(1, 4, −3) and R(2, −2, 5). Find the size of angle PQR, to 3 significant figures.

Find the value of t for which the two vectors are perpendicular.

1. (a) \(\mathbf{a} = \begin{pmatrix}2t\\-3\\1 - t\end{pmatrix},\ \mathbf{b} = \begin{pmatrix}3\\2\\1\end{pmatrix}\)
2. (b) \(\mathbf{a} = (-1 + 2t)\mathbf{i} + 3\mathbf{j} + t\mathbf{k},\ \mathbf{b} = 3\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}\)

The vectors \(\mathbf{a} = \begin{pmatrix}t\\4\\2 + t\end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix}-4 + t\\2\\-1\end{pmatrix}\) are perpendicular. Find the possible values of \(t\).

A crane has its hook point at A(0, 0, 15). Two ropes run from A to the ground at B(6, −4, 0) and C(−3, −7, 0). Find the angle between the two ropes, to 3 significant figures. Distances are in metres.

A tow rope pulls a barge with the force \(\mathbf{F} = \begin{pmatrix}5\\3t\\-2 + t\end{pmatrix}\) N, where \(t\) is a constant. The object moves through the displacement \(\mathbf{d} = \begin{pmatrix}-4\\2\\2\end{pmatrix}\) m. The work done is zero. Find the value of \(t\).

The points P, Q and R have coordinates P(0, 2, −1), Q(3, 2, 3) and R(−3, 5, 2).

1. (a) Find the size of angle QPR.
2. (b) Hence find the area of triangle PQR.

Use the scalar product to show that \(\mathbf{p} = \begin{pmatrix}4\\-2\\6\end{pmatrix}\) and \(\mathbf{q} = \begin{pmatrix}10\\-5\\15\end{pmatrix}\) are parallel, and say whether they point in the same direction.