Cross products and unit circle

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I am first year maths student at university.

I am currently studying cross products.

For this question,

I managed to solve the first 2 parts but am struggling with the general question at the end.

An explanation on how to solve the last one will be greatly appreciated.

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2 Answers

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Hint: Recall definition of cross product $\mathbf{a}\times\mathbf{b}=ab\sin(\theta)\,\mathbf{n}$. So $\mathbf{p}\times\mathbf{q}=\sin(\angle POQ)\,\mathbf{n}$ and you know $\mathbf{n}=\pm\mathbf{k}$...

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Hints:

• You are told $|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}| \, |\mathbf{b}| \,|\sin(\theta)|$ so you can put bounds on $r$
• Your answers to the first part may suggest whether these bounds are achieved

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