Let $$\overrightarrow A = 2\overrightarrow i + \overrightarrow k ,\,\overrightarrow B = \overrightarrow i + \overrightarrow j + \overrightarrow k ,$$ and $$\overrightarrow C = 4\overrightarrow i - 3\overrightarrow j + 7\overrightarrow k .$$ Determine a vector $$\overrightarrow R .$$ Satisfying $$\overrightarrow R \times \overrightarrow B = \overrightarrow C \times \overrightarrow B $$ and $$\overrightarrow R \,.\,\overrightarrow A = 0$$

If vectors $$\overrightarrow A ,\overrightarrow B ,\overrightarrow C $$ are coplanar, show that $$$\left| {\matrix{ {} & {\overrightarrow {a.} } & {} & {\overrightarrow {b.} } & {} & {\overrightarrow {c.} } \cr {\overrightarrow {a.} } & {\overrightarrow {a.} } & {\overrightarrow {a.} } & {\overrightarrow {b.} } & {\overrightarrow {a.} } & {\overrightarrow {c.} } \cr {\overrightarrow {b.} } & {\overrightarrow {a.} } & {\overrightarrow {b.} } & {\overrightarrow {b.} } & {\overrightarrow {b.} } & {\overrightarrow {c.} } \cr } } \right| = \overrightarrow 0 $$$

In a triangle $$OAB,E$$ is the midpoint of $$BO$$ and $$D$$ is a point on $$AB$$ such that $$AD:DB=2:1.$$ If $$OD$$ and $$AE$$ intersect at $$P,$$ determine the ratio $$OP:PD$$ using vector methods.

Let $$OA$$ $$CB$$ be a parallelogram with $$O$$ at the origin and $$OC$$ a diagonal. Let $$D$$ be the midpoint of $$OA.$$ Using vector methods prove that $$BD$$ and $$CO$$ intersect in the same ratio. Determine this ratio.