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Let \(\mathbf{r}\) be the position vector of a point in three-dimensional space. Describe fully the locus of the point whose position vector is \(\mathbf{r}\) in each of the following four cases:

1. \(\left(\mathbf{a-b}\right) \cdot \mathbf{r}=\frac{1}{2}(\left|\mathbf{a}\right|^{2}-\left|\mathbf{b}\right|^{2});\)
2. \(\left(\mathbf{a-r}\right)\cdot\left(\mathbf{b-r}\right)=0;\)
3. \(\left|\mathbf{r-a}\right|^{2}=\frac{1}{2}\left|\mathbf{a-b}\right|^{2};\)
4. \(\left|\mathbf{r-b}\right|^{2}=\frac{1}{2}\left|\mathbf{a-b}\right|^{2}.\)

Prove algebraically that the equations \((i)\) and \((ii)\) together are equivalent to \((iii)\) and \((iv)\) together. Explain carefully the geometrical meaning of this equivalence.

The sides of a triangle have lengths \(p-q\), \(p\) and \(p+q\), where \(p>q> 0\,\). The largest and smallest angles of the triangle are \(\alpha\) and \(\beta\), respectively. Show by means of the cosine rule that \[ 4(1-\cos\alpha)(1-\cos\beta) = \cos\alpha + \cos\beta \,. \] In the case \(\alpha = 2\beta\), show that \(\cos\beta=\frac34\) and hence find the ratio of the lengths of the sides of the triangle.