Show, by vector methods, that the angular bisectors of a triangle are concurrent and find an expression for the position vector of the point of concurrency in terms of the position vectors of the vertices.

Find $$3-$$dimensional vectors $${\overrightarrow v _1},{\overrightarrow v _2},{\overrightarrow v _3}$$ satisfying
$$\,{\overrightarrow v _1}.{\overrightarrow v _1} = 4,\,{\overrightarrow v _1}.{\overrightarrow v _2} = - 2,\,{\overrightarrow v _1}.{\overrightarrow v _3} = 6,\,\,{\overrightarrow v _2}.{\overrightarrow v _2}$$
$$ = 2,\,{\overrightarrow v _2}.{\overrightarrow v _3} = - 5,\,{\overrightarrow v _3}.{\overrightarrow v _3} = 29$$

Let $$u$$ and $$v$$ be units vectors. If $$w$$ is a vector such that $$w + \left( {w \times u} \right) = v,$$ then prove that $$\left| {\left( {u \times v} \right) \cdot w} \right| \le 1/2$$ and that the equality holds if and only if $$u$$ is perpendicular to $$v .$$

Prove, by vector methods or otherwise, that the point of intersection of the diagonals of a trapezium lies on the line passing through the mid-points of the parallel sides. (You may assume that the trapezium is not a parallelogram.)