For what values of $\lambda$ are $\vec{a}$ and $\vec{c}$ unit collinear vectors,and given $|\vec{b}| = 6$,if $\vec{b} - 3\vec{c} = \lambda \vec{a}$,then $\lambda = ......$

In the given figure,if a vector $x$ satisfies the equation $x - w = v$,then $x = ?$

If $\vec{a}, \vec{b}$ and $\vec{c}$ are non-coplanar vectors and if $\vec{d}$ is such that $\vec{d} = \frac{1}{x}(\vec{a} + \vec{b} + \vec{c})$ and $\vec{d} = \frac{1}{y}(\vec{b} + \vec{c} + \vec{d})$ where $x$ and $y$ are non-zero real numbers, then $\frac{1}{xy}(\vec{a} + \vec{b} + \vec{c} + \vec{d})$ is equal to:

In a quadrilateral $ABCD$, the point $P$ divides $DC$ in the ratio $1:2$ and $Q$ is the midpoint of $AC$. If $\overrightarrow{AB}+2\overrightarrow{AD}+\overrightarrow{BC}-2\overrightarrow{DC}=k\overrightarrow{PQ}$, then $k$ is equal to

Let $\triangle PQR$ be a triangle. Let $\vec{a}=\overline{QR}, \vec{b}=\overline{RP}$ and $\vec{c}=\overline{PQ}$. If $|\vec{a}|=12, |\vec{b}|=4\sqrt{3}$ and $\vec{b} \cdot \vec{c}=24$,then which of the following is (are) true?
$(A) \frac{|\vec{c}|^2}{2}-|\vec{a}|=12$
$(B) \frac{|\vec{c}|^2}{2}+|\vec{a}|=30$
$(C) |\vec{a} \times \vec{b}+\vec{c} \times \vec{a}|=48\sqrt{3}$
$(D) \vec{a} \cdot \vec{b}=-72$

Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that $|\vec{a} + \vec{b}| = \sqrt{3}$. If $\vec{c} = \vec{a} + 2\vec{b} + 3(\vec{a} \times \vec{b})$,then $2|\vec{c}|$ is equal to