The value of $\hat{i} \cdot(\hat{j} \times \hat{k})+\hat{j} \cdot(\hat{i} \times \hat{k})+\hat{k} \cdot(\hat{i} \times \hat{j})$ is . . . . . . .

Find the angle between the lines whose direction ratios are $a, b, c$ and $b-c, c-a, a-b$. (in $^{\circ}$)

Let $u = -2 \hat{i} + 2 \hat{j} + \hat{k}$ and $v = \hat{i} - 2 \hat{j} + 2 \hat{k}$. Then the component of $v$ on $u$ is

The vector $a + b$ bisects the angle between the vectors $a$ and $b$,if

In a triangle $ABC$,if $|\overrightarrow{BC}|=3$,$|\overrightarrow{AC}|=5$,and $|\overrightarrow{BA}|=7$,then the projection of the vector $\overrightarrow{BA}$ on $\overrightarrow{BC}$ is equal to:

Let $a, b$ and $c$ be vectors with magnitudes $3, 4$ and $5$ respectively and $a + b + c = 0$. Then the value of $a \cdot b + b \cdot c + c \cdot a$ is: