The scalar product Problem set A

The diagram shows two vectors \(\mathbf{a}\) and \(\mathbf{b}\) that leave the same point, with \(|\mathbf{a}|\) = 7, \(|\mathbf{b}|\) = 12 and an angle of 40° between them. Find \(\mathbf{a}\cdot\mathbf{b}\), to 3 significant figures.

The diagram shows two vectors \(\mathbf{a}\) and \(\mathbf{b}\) that leave the same point, with \(|\mathbf{a}|\) = 10, \(|\mathbf{b}|\) = 6 and an angle of 150° between them. Find \(\mathbf{a}\cdot\mathbf{b}\), to 3 significant figures.

Find the scalar product of each pair of vectors.

1. (a) \(\mathbf{a} = \begin{pmatrix}2\\3\\1\end{pmatrix},\ \mathbf{b} = \begin{pmatrix}4\\-1\\5\end{pmatrix}\)
2. (b) \(\mathbf{a} = \begin{pmatrix}-3\\0\\6\end{pmatrix},\ \mathbf{b} = \begin{pmatrix}5\\7\\-2\end{pmatrix}\)
3. (c) \(\mathbf{a} = \begin{pmatrix}1\\-4\\2\end{pmatrix},\ \mathbf{b} = \begin{pmatrix}8\\1\\-2\end{pmatrix}\)

Find the scalar product of each pair of vectors.

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

A crane applies a force \(\mathbf{F} = \begin{pmatrix}15\\-6\\4\end{pmatrix}\) N to a load, and the load moves through the displacement \(\mathbf{d} = \begin{pmatrix}2\\5\\3\end{pmatrix}\) m. The work done is \(\mathbf{F}\cdot\mathbf{d}\) joules. Find the work done.

A rope pulls a sled with a force of 85 N. The rope makes an angle of 30° with the ground and the sled moves 40 m along the ground. The work done is the scalar product of the force and the displacement. Find the work done, in joules, to 3 significant figures.

The points A, B and C have coordinates A(1, 2, −1), B(4, −2, 3) and C(2, 5, 0). Find \(\overrightarrow{AB}\cdot\overrightarrow{AC}\).

Let \(\mathbf{a} = \begin{pmatrix}2\\-1\\3\end{pmatrix}\), \(\mathbf{b} = \begin{pmatrix}4\\0\\-2\end{pmatrix}\) and \(\mathbf{c} = \begin{pmatrix}-1\\5\\1\end{pmatrix}\).

1. (a) Find \(\mathbf{a}\cdot\mathbf{b}\) and \(\mathbf{a}\cdot\mathbf{c}\).
2. (b) Find \(\mathbf{b}+\mathbf{c}\), and then \(\mathbf{a}\cdot(\mathbf{b}+\mathbf{c})\).
3. (c) Show that \(\mathbf{a}\cdot(\mathbf{b}+\mathbf{c}) = \mathbf{a}\cdot\mathbf{b} + \mathbf{a}\cdot\mathbf{c}\).