Let $a, b$ and $c$ be three non-coplanar vectors and let $p, q$ and $r$ be the vectors defined by $p=\frac{b \times c}{[a b c]}, q=\frac{c \times a}{[a b c]}, r=\frac{a \times b}{[a b c]}$. Then,$(a+b) \cdot p+(b+c) \cdot q+(c+a) \cdot r$ is equal to

The volume of the parallelepiped formed by the vectors $\hat{i} + m \hat{j} + \hat{k}$,$\hat{j} + m \hat{k}$,and $m \hat{i} + \hat{k}$ becomes minimum when $m$ is

If the vectors $\vec{a}=\lambda \hat{i}+\mu \hat{j}+4 \hat{k}$,$\vec{b}=2 \hat{i}+4 \hat{j}-2 \hat{k}$ and $\vec{c}=2 \hat{i}+3 \hat{j}+\hat{k}$ are coplanar and the projection of $\vec{a}$ on the vector $\vec{b}$ is $\sqrt{54}$ units,then the sum of all possible values of $\lambda+\mu$ is equal to:

Let $a = \hat{i} - 2\hat{j} + 3\hat{k}$ and $b = 2\hat{i} + \hat{j} + \hat{k}$. If $c$ is a unit vector such that $[a \ b \ c]$ is maximum,then $c =$

Let $\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$ and $\vec{b}=\hat{i}+\hat{j}-\hat{k}$. If $\vec{c}$ is a vector such that $\vec{a} \cdot \vec{c}=11$,$\vec{b} \cdot(\vec{a} \times \vec{c})=27$ and $\vec{b} \cdot \vec{c}=-\sqrt{3}|\vec{b}|$,then $|\vec{a} \times \vec{c}|^2$ is equal to

If $d = \lambda (a \times b) + \mu (b \times c) + \nu (c \times a)$ and $[a, b, c] = \frac{1}{8}$,then $\lambda + \mu + \nu$ is equal to