The vector product Problem set A

The vectors \(\mathbf{a}\) and \(\mathbf{b}\) have magnitudes 11 and 6. The angle between them is \(65^\circ\). Find \(|\mathbf{a} \times \mathbf{b}|\).

The vectors \(\mathbf{a}\) and \(\mathbf{b}\) have magnitudes 5 and 12. The angle between them is \(140^\circ\). Find \(|\mathbf{a} \times \mathbf{b}|\).

Find \(\mathbf{a} \times \mathbf{b}\) when \(\mathbf{a} = \begin{pmatrix}1\\4\\-2\end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix}3\\-1\\2\end{pmatrix}\)

Find \(\mathbf{a} \times \mathbf{b}\) when \(\mathbf{a} = \begin{pmatrix}0\\3\\-1\end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix}2\\-4\\5\end{pmatrix}\)

Find \(\mathbf{a} \times \mathbf{b}\) when \(\mathbf{a} = 2\mathbf{i} + 3\mathbf{j} - \mathbf{k}\) and \(\mathbf{b} = \mathbf{i} - 2\mathbf{j} + 4\mathbf{k}\)

Find \(\mathbf{a} \times \mathbf{b}\) when \(\mathbf{a} = 5\mathbf{i} - 2\mathbf{k}\) and \(\mathbf{b} = \mathbf{i} + 3\mathbf{j}\)

A mechanic pulls on a spanner. The spanner is 0.35 m long and the pull is a force of 80 N at \(70^\circ\) to the spanner. The turning effect (the torque) is the magnitude of the vector product of the position vector and the force. Find the torque, in N m.

A triangular roof panel has two edges from one corner given, in metres, by the vectors \(\mathbf{a} = \begin{pmatrix}4\\0\\1\end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix}0\\3\\1\end{pmatrix}\). Find a vector that is perpendicular to the panel.

The vectors are \(\mathbf{a} = \begin{pmatrix}2\\5\\-3\end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix}1\\-2\\4\end{pmatrix}\).

1. (a) Find \(\mathbf{a} \times \mathbf{b}\).
2. (b) Show that \(\mathbf{a} \times \mathbf{b}\) is perpendicular to \(\mathbf{a}\).