The vector product Problem set B

The vectors \(\mathbf{a}\) and \(\mathbf{b}\) have magnitudes 7 and 8. The angle between them is \(52^\circ\). Find \(|\mathbf{a} \times \mathbf{b}|\).

The vectors \(\mathbf{a}\) and \(\mathbf{b}\) have magnitudes 10 and 3. The angle between them is \(125^\circ\). Find \(|\mathbf{a} \times \mathbf{b}|\).

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

Find \(\mathbf{a} \times \mathbf{b}\) when \(\mathbf{a} = \begin{pmatrix}4\\-2\\0\end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix}-3\\1\\6\end{pmatrix}\)

Find \(\mathbf{a} \times \mathbf{b}\) when \(\mathbf{a} = \mathbf{i} + 4\mathbf{j} - 3\mathbf{k}\) and \(\mathbf{b} = 2\mathbf{i} - \mathbf{j} + 5\mathbf{k}\)

Find \(\mathbf{a} \times \mathbf{b}\) when \(\mathbf{a} = -2\mathbf{i} + 4\mathbf{j}\) and \(\mathbf{b} = \mathbf{j} + 6\mathbf{k}\)

A cyclist pushes a pedal. The crank is 0.17 m long and the push is a force of 140 N at \(50^\circ\) to the crank. 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 sail has two edges from one corner given, in metres, by the vectors \(\mathbf{a} = \begin{pmatrix}5\\0\\2\end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix}0\\4\\3\end{pmatrix}\). Find a vector that is perpendicular to the sail.

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

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