$A$ unit vector perpendicular to vector $c$ and coplanar with vectors $a$ and $b$ is

$a \times [a \times (a \times b)]$ is equal to

Let $a, b, c$ be three vectors. Then $a \times (b \times c) = (a \times b) \times c$ if:

Let $\vec{x}, \vec{y}$ and $\vec{z}$ be three vectors each of magnitude $\sqrt{2}$ and the angle between each pair of them is $\frac{\pi}{3}$. If $\vec{a}$ is a nonzero vector perpendicular to $\vec{x}$ and $\vec{y} \times \vec{z}$ and $\vec{b}$ is a nonzero vector perpendicular to $\vec{y}$ and $\vec{z} \times \vec{x}$,then
$(A)$ $\vec{b}=(\vec{b} \cdot \vec{z})(\vec{z}-\vec{x})$
$(B)$ $\vec{a}=(\vec{a} \cdot \vec{y})(\vec{y}-\vec{z})$
$(C)$ $\vec{a} \cdot \vec{b}=-(\vec{a} \cdot \vec{y})(\vec{b} \cdot \vec{z})$
$(D)$ $\vec{a}=(\vec{a} \cdot \vec{y})(\vec{z}-\vec{y})$

Let $\vec{a}$ be a unit vector and $\vec{b}$ be a nonzero vector not parallel to $\vec{a}$. The angles of the triangle,two of whose sides are represented by $\sqrt{3}(\vec{a} \times \vec{b})$ and $\vec{b} - (\vec{a} \cdot \vec{b})\vec{a}$,are

If $\vec{a}$ and $\vec{b}$ are two non-collinear vectors, then $\frac{\vec{a} \times(\vec{b} \times \vec{a})}{|\vec{a}|^2}$ represents: