RI/2021/JC1/Promo/P1/Q03

The points $A, B$ and $C$ have position vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ respectively.
(a) Show that the area of triangle $A B C$ is $\frac{1}{2}|\mathbf{a} \times \mathbf{b}+\mathbf{b} \times \mathbf{c}+\mathbf{c} \times \mathbf{a}|$. Hence show that the shortest distance from $B$ to $A C$ is
$$\frac{|\mathbf{a} \times \mathbf{b}+\mathbf{b} \times \mathbf{c}+\mathbf{c} \times \mathbf{a}|}{|\mathbf{c}-\mathbf{a}|}$$

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(b) Given that $\mathbf{a}$ and $\mathbf{b}$ are non-zero vectors such that $|\mathbf{a}-\mathbf{b}|=|\mathbf{a}+\mathbf{b}|$, find the value of $\mathbf{a} \cdot \mathbf{b}$.

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