The vector product Problem set B One question at a time

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

• \(|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta = 7 \times 8 \times \sin 52^\circ\)
• \(= 44.1\)

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

• \(|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta = 10 \times 3 \times \sin 125^\circ\)
• The angle is obtuse. \(\sin 125^\circ = \sin 55^\circ\), so the calculator gives the same positive value
• \(= 24.6\)

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

• \(\mathbf{a} \times \mathbf{b} = \begin{pmatrix}(-3)(4) - (1)(2)\\ (1)(-1) - (2)(4)\\ (2)(2) - (-3)(-1)\end{pmatrix}\)
• \(= \begin{pmatrix}-14\\-9\\1\end{pmatrix}\)

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

• \(\mathbf{a} \times \mathbf{b} = \begin{pmatrix}(-2)(6) - (0)(1)\\ (0)(-3) - (4)(6)\\ (4)(1) - (-2)(-3)\end{pmatrix}\)
• \(= \begin{pmatrix}-12\\-24\\-2\end{pmatrix}\)

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

• \(\mathbf{a} = \begin{pmatrix}1\\4\\-3\end{pmatrix},\ \mathbf{b} = \begin{pmatrix}2\\-1\\5\end{pmatrix}\)
• \(\mathbf{a} \times \mathbf{b} = \begin{pmatrix}(4)(5) - (-3)(-1)\\ (-3)(2) - (1)(5)\\ (1)(-1) - (4)(2)\end{pmatrix}\)
• \(= \begin{pmatrix}17\\-11\\-9\end{pmatrix}\)
• \(= 17\mathbf{i} - 11\mathbf{j} - 9\mathbf{k}\)

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

• \(\mathbf{a} = \begin{pmatrix}-2\\4\\0\end{pmatrix},\ \mathbf{b} = \begin{pmatrix}0\\1\\6\end{pmatrix}\)
• \(\mathbf{a} \times \mathbf{b} = \begin{pmatrix}(4)(6) - (0)(1)\\ (0)(0) - (-2)(6)\\ (-2)(1) - (4)(0)\end{pmatrix}\)
• \(= \begin{pmatrix}24\\12\\-2\end{pmatrix}\)
• \(= 24\mathbf{i} + 12\mathbf{j} - 2\mathbf{k}\)

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.

• \(|\mathbf{r} \times \mathbf{F}| = |\mathbf{r}||\mathbf{F}|\sin\theta = 0.17 \times 140 \times \sin 50^\circ\)
• \(= 18.2\)

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.

• \(\mathbf{a} \times \mathbf{b} = \begin{pmatrix}(0)(3) - (2)(4)\\ (2)(0) - (5)(3)\\ (5)(4) - (0)(0)\end{pmatrix}\)
• \(= \begin{pmatrix}-8\\-15\\20\end{pmatrix}\)

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 \(\mathbf{a} \times \mathbf{b}\).

   • \(\mathbf{a} \times \mathbf{b} = \begin{pmatrix}(1)(2) - (-2)(4)\\ (-2)(-1) - (3)(2)\\ (3)(4) - (1)(-1)\end{pmatrix}\)
   • \(= \begin{pmatrix}10\\-4\\13\end{pmatrix}\)

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

   • \(\mathbf{a} \cdot (\mathbf{a} \times \mathbf{b}) = (3)(10) + (1)(-4) + (-2)(13) = 0\)
   • The scalar product is 0, so the vectors are perpendicular