3D vectors: lines and planes

Vector playground

The points $O$, $A$ and $B$ are on a plane such that relative to the point $O$, the points $A$ and $B$ have non-parallel position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively.
The point $C$ with position vector $\mathbf{c}$ is on the plane $OAB$ such that $OC$ bisects the angle $AOB$.
Show that $\left( \frac{\mathbf{a}}{\left| \mathbf{a} \right|}-\frac{\mathbf{b}}{\left| \mathbf{b} \right|} \right)\cdot \mathbf{c}=0$.[2]
The lines $AB$ and $OC$ intersect at $P$. By first verifying that $\overrightarrow{OC}$ is parallel to $\frac{\mathbf{a}}{\left| \mathbf{a} \right|}+\frac{\mathbf{b}}{\left| \mathbf{b} \right|}$, show that the ratio of $AP:PB=\left| \mathbf{a} \right|:\left| \mathbf{b} \right|$.[6]

The points $O$, $A$ and $B$ are on a plane such that relative to the point $O$, the points $A$ and $B$ have non-parallel position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively.

The point $C$ with position vector $\mathbf{c}$ is on the plane $OAB$ such that $OC$ bisects the angle $AOB$.

Show that $\left( \frac{\mathbf{a}}{\left| \mathbf{a} \right|}-\frac{\mathbf{b}}{\left| \mathbf{b} \right|} \right)\cdot \mathbf{c}=0$.

The lines $AB$ and $OC$ intersect at $P$. By first verifying that $\overrightarrow{OC}$ is parallel to $\frac{\mathbf{a}}{\left| \mathbf{a} \right|}+\frac{\mathbf{b}}{\left| \mathbf{b} \right|}$, show that the ratio of $AP:PB=\left| \mathbf{a} \right|:\left| \mathbf{b} \right|$.

2017 ACJC P1 Q5

2017 ACJC P1 Q5

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