In a triangle $ABC$,if $\overline{BC} = \bar{i} - 2\bar{j} + 2\bar{k}$ and $\overline{CA} = 6\bar{i} + 3\bar{j} - 2\bar{k}$,then the perimeter of the triangle is

If $\vec{a}$ is a nonzero vector of magnitude $a$ and $\lambda$ is a nonzero scalar,then $\lambda \vec{a}$ is a unit vector if

Let $\vec{a}=3 \hat{i}+\hat{j}-2 \hat{k}$,$\vec{b}=-5 \hat{i}+7 \hat{j}$,and $\vec{c}=3 \hat{i}+y \hat{j}$ be three vectors such that $|\vec{a}-\vec{b}+\vec{c}|=\sqrt{141}$. If $y_1$ and $y_2$ are the values of $y$ satisfying the given condition,then $|y_1-y_2|=$

If $a = (1, -1)$ and $b = (-2, m)$ are two collinear vectors,then $m = $

$(a \cdot i)i + (a \cdot j)j + (a \cdot k)k = $

If $a = \hat{i} + 4 \hat{j}$,$b = 2 \hat{i} - 2 \hat{j}$,and $c = 5 \hat{i} + 9 \hat{j}$,then $c$ is equal to: