Exploring Cross Product Properties

00:00 6 marks

Let \(\mathbf{a} = (1, 2, 3)\), \(\mathbf{b} = (0, -1, 2)\), and \(\mathbf{c} = (3, 0, -1)\).

(a) Compute the cross product \(\mathbf{a} \times \mathbf{b}\) and verify that \(\mathbf{a} \times \mathbf{b} = – (\mathbf{b} \times \mathbf{a})\).

(b) Demonstrate the distributive property by first computing \(\mathbf{a} \times (\mathbf{b} + \mathbf{c})\) and then verifying that it equals \(\mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c}\).

Hint Hint

Hint

Remember that the cross product is anticommutative and distributes over vector addition.

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\(\mathbf{a} \times \mathbf{b} = (7, -2, -1)\), \(\mathbf{b} \times \mathbf{a} = (-7, 2, 1)\), and \(\mathbf{a} \times (\mathbf{b} + \mathbf{c}) = (5, 8, -7)\) which equals \(\mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c} = (7, -2, -1) + (-2, 10, -6) = (5, 8, -7)\).

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