Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{b}|=1$ and $|\vec{b} \times \vec{a}|=2$. Then $|(\vec{b} \times \vec{a})-\vec{b}|^2$ is equal to

$a \times (b \times c)$ is coplanar with

Let $\vec{a}=2 \hat{i}+\alpha \hat{j}+\hat{k}$,$\vec{b}=-\hat{i}+\hat{k}$,and $\vec{c}=\beta \hat{j}-\hat{k}$,where $\alpha$ and $\beta$ are integers and $\alpha \beta=-6$. Let the values of the ordered pair $(\alpha, \beta)$ for which the area of the parallelogram with diagonals $\vec{a}+\vec{b}$ and $\vec{b}+\vec{c}$ is $\frac{\sqrt{21}}{2}$,be $(\alpha_1, \beta_1)$ and $(\alpha_2, \beta_2)$. Then $\alpha_1^2+\beta_1^2-\alpha_2 \beta_2$ is equal to

For any two vectors $a$ and $b$,if $a \times b = 0$,then

$a, b, c, d$ are coplanar vectors, then $(a \times b) \times (c \times d)$ is equal to

Let vectors $a, b, c$ and $d$ be such that $(a \times b) \times (c \times d) = 0$. If $a$ and $b$ lie in plane $P_1$ and $c$ and $d$ lie in plane $P_2$,find the angle between $P_1$ and $P_2$.