The scalar product Problem set B

The diagram shows two vectors \(\mathbf{a}\) and \(\mathbf{b}\) that leave the same point, with \(|\mathbf{a}|\) = 15, \(|\mathbf{b}|\) = 4 and an angle of 65° 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 \(|\mathbf{a}|\) = 5, \(|\mathbf{b}|\) = 14 and an angle of 110° between them. Find \(\mathbf{a}\cdot\mathbf{b}\), to 3 significant figures.

Find the scalar product of each pair of vectors.

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

Find the scalar product of each pair of vectors.

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

A tug applies a force \(\mathbf{F} = \begin{pmatrix}-8\\9\\5\end{pmatrix}\) N to a load, and the load moves through the displacement \(\mathbf{d} = \begin{pmatrix}4\\1\\6\end{pmatrix}\) m. The work done is \(\mathbf{F}\cdot\mathbf{d}\) joules. Find the work done.

A handle pulls a trolley with a force of 18 N. The handle makes an angle of 52° with the floor and the trolley moves 9 m along the floor. 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 P, Q and R have coordinates P(−2, 0, 3), Q(1, 4, −1) and R(3, −3, 2). Find \(\overrightarrow{PQ}\cdot\overrightarrow{PR}\).

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

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