Let $\overrightarrow a$, $\overrightarrow b$ and $\overrightarrow c$ be three non-zero non-coplanar vectors. Let the position vectors of four points $A,B,C$ and $D$ be $$\overrightarrow a - \overrightarrow b + \overrightarrow c ,\lambda \overrightarrow a - 3\overrightarrow b + 4\overrightarrow c , - \overrightarrow a + 2\overrightarrow b - 3\overrightarrow c $$ and $2\overrightarrow a - 4\overrightarrow b + 6\overrightarrow c$ respectively. If $\overrightarrow {AB} ,\overrightarrow {AC}$ and $\overrightarrow {AD}$ are coplanar, then $\lambda$ is equal to __________.

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}+\vec{b}|^{2}=|\vec{a}|^{2}+2|\vec{b}|^{2}, \vec{a} \cdot \vec{b}=3$ and…

If $\left( {\overrightarrow a + 3\overrightarrow b } \right)$ is perpendicular to $\left( {7\overrightarrow a - 5\overrightarrow b }…

The least positive integral value of $\alpha$, for which the angle between the vectors $\alpha \hat{i}-2 \hat{j}+2 \hat{k}$ and $\alpha…

Let $\overrightarrow a$ = $\widehat i$ + 2$\widehat j$ $-$ 3$\widehat k$ and $\overrightarrow b = 2\widehat i$ $-$ 3$\widehat j$ +…

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}|=1,|\vec{b}|=4$, and $\vec{a} \cdot \vec{b}=2$. If $\vec{c}=(2 \vec{a}…