Vectors Questions

$ABC$ is a triangle, right-angled at $A$. The resultant of the forces acting along $AB$ and $AC$ with magnitudes $\frac{1}{AB}$ and $\frac{1}{AC}$ respectively is a force along $AD$, where $D$ is the foot of the perpendicular from $A$ onto $BC$. The magnitude of the resultant is:

If $(\vec{a}\times \vec{b})\times \vec{c}=\vec{a}\times (\vec{b}\times \vec{c})$, where $\vec{a},\vec{b}$ and $\vec{c}$ are any three vectors such that $\vec{a}\cdot \vec{b}\ne 0$ and $\vec{b}\cdot \vec{c}\ne 0$, then $\vec{a}$ and $\vec{c}$ are:

A particle has two velocities of equal magnitude $u$ inclined to each other at an angle $\theta$. If one of them is halved, the angle between the other and the original resultant velocity is bisected by the new resultant. Then $\theta$ is:

The values of $a$ for which the points $A,B,C$ with position vectors $2\hat{i}-\hat{j}+\hat{k}$, $\hat{i}-3\hat{j}-5\hat{k}$ and $a\hat{i}-3\hat{j}+\hat{k}$ respectively are the vertices of a right-angled triangle with $\angle C=\pi /2$ are: