Angles, perpendicular and parallel vectors Problem set A

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

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

The points A, B and C have coordinates A(1, −2, 4), B(3, 1, −1) and C(−2, 4, 2). Find the size of angle ABC, to 3 significant figures.

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

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

The vectors \(\mathbf{a} = \begin{pmatrix}2t\\1\\t\end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix}-3 + t\\-6\\2\end{pmatrix}\) are perpendicular. Find the possible values of \(t\).

A radio mast has its top at A(0, 0, 12). Two cables run from A to anchor points B(8, 3, 0) and C(−5, 6, 0) on the ground. Find the angle between the two cables, to 3 significant figures. Distances are in metres.

A winch pulls a block with the force \(\mathbf{F} = \begin{pmatrix}2t\\1 + t\\3\end{pmatrix}\) N, where \(t\) is a constant. The object moves through the displacement \(\mathbf{d} = \begin{pmatrix}2\\-6\\3\end{pmatrix}\) m. The work done is zero. Find the value of \(t\).

The points A, B and C have coordinates A(1, 0, 2), B(4, 4, 2) and C(−2, 3, 6).

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

Use the scalar product to show that \(\mathbf{p} = \begin{pmatrix}2\\-6\\4\end{pmatrix}\) and \(\mathbf{q} = \begin{pmatrix}-3\\9\\-6\end{pmatrix}\) are parallel, and say whether they point in the same direction.