Vector Cross Product and Angle

00:00 7 marks

Consider the vectors \(\boldsymbol{a} = \boldsymbol{i} – 2\boldsymbol{j} + 3\boldsymbol{k}\) and \(\boldsymbol{b} = 2\boldsymbol{i} + \boldsymbol{j} – \boldsymbol{k}\).

(a) Find \(\boldsymbol{a} \times \boldsymbol{b}\). [3]
(b) Find the sine of the angle between \(\boldsymbol{a}\) and \(\boldsymbol{b}\). [4]

Hint Hint

Hint

For (a), set up the 3×3 determinant with unit vectors in the first row and the components of \(\boldsymbol{a}\) and \(\boldsymbol{b}\) in the following rows. For (b), use the relationship \(|\boldsymbol{a} \times \boldsymbol{b}| = |\boldsymbol{a}||\boldsymbol{b}|\sin \theta\) to solve for \(\sin \theta\).

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Answer Answer

Complete Answer

Part (a): \(\boldsymbol{a} \times \boldsymbol{b} = -\boldsymbol{i} + 7\boldsymbol{j} + 5\boldsymbol{k}\).
Part (b): \(\sin \theta = \frac{5\sqrt{3}}{\sqrt{14}\sqrt{6}} = \frac{5}{2\sqrt{7}}\).

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