If $\overline{a}=2 \hat{i}+2 \hat{j}+3 \hat{k}$,$\overline{b}=-\hat{i}+2 \hat{j}+\hat{k}$ and $\overline{c}=3 \hat{i}+\hat{j}$ such that $\overline{b}+\lambda \overline{a}$ is perpendicular to $\overline{c}$,then $\lambda$ is

If $\vec{a} = \hat{i} + 3\hat{j} - 2\hat{k}$ and $\vec{b} = 4\hat{i} - 2\hat{j} + 4\hat{k}$,then $(2\vec{a} + \vec{b}) \cdot (\vec{a} - 2\vec{b}) = \dots$

If three consecutive vertices of a parallelogram are $A(4,3,5)$,$B(0,6,0)$,$C(-8,1,4)$ and $D$ is the fourth vertex,then the angle between $AC$ and $BD$ is

Let $\vec{a}=\hat{i}+4 \hat{j}+2 \hat{k}, \vec{b}=3 \hat{i}-2 \hat{j}+7 \hat{k}$ and $\vec{c}=2 \hat{i}-\hat{j}+4 \hat{k}$. If a vector $\vec{d}$ satisfies $\vec{d} \times \vec{b}=\vec{c} \times \vec{b}$ and $\vec{d} \cdot \vec{a}=24$,then $|\vec{d}|^2$ is equal to $.........$.

If $\vec{a} = 2\hat{i} + 3\hat{j} + \hat{k}$,$\vec{b} = \hat{i} - \hat{j} + \hat{k}$,$\vec{c} = \hat{i} + \hat{j} + \hat{k}$ and let $\vec{d}$ be such that $\vec{a} \times \vec{b} = \vec{d} \times \vec{b}$ and $\vec{d} \cdot \vec{c} = 8$,then the value of $\vec{d} \cdot \vec{b}$ is:

Let $ABC$ be an equilateral triangle of side $a$. $M$ and $N$ are two points on the sides $AB$ and $AC$ respectively such that $\overrightarrow{AN} = K \overrightarrow{AC}$ and $\overrightarrow{AB} = 3 \overrightarrow{AM}$. If the vectors $\overrightarrow{BN}$ and $\overrightarrow{CM}$ are perpendicular,then $K =$