The projection of the vector $i+j+k$ along the vector $j$ is

If $P(1, 3, -7)$ and $Q(5, -2, 4)$,then $|\vec{PQ}| = \dots$

$ABCDE$ is a pentagon. Forces $\overline{AB}, \overline{AE}, \overline{DC}, \overline{ED}$ act at a point. Which force should be added to this system to make the resultant $2\overline{AC}$?

If $a, b, c$ are three linearly independent vectors and there exists a non-zero scalar triad $(l, m, n)$ such that $l(3a + 2b + c) + m(2a + 2b + 3c) + n(a + 2b + 5c) = 0$,then:

Let $OA = a, OB = b$ be two non-collinear vectors,$OP = x_1 a + y_1 b, OQ = x_2 a + y_2 b$ and $A^{\prime}O = OA, B^{\prime}O = OB$. If $x_1 = -\frac{3}{4}, x_2 = \frac{1}{3}, y_1 = \frac{7}{4}, y_2 = \frac{5}{3}$,then

Let the three sides of a triangle $ABC$ be represented by the vectors $\vec{AB} = 2\hat{i}-\hat{j}+\hat{k}$,$\vec{BC} = 3\hat{i}-4\hat{j}-4\hat{k}$,and $\vec{CA} = \hat{i}-3\hat{j}-5\hat{k}$. Let $G$ be the centroid of the triangle $ABC$. Then $6(|\overrightarrow{AG}|^2+|\overrightarrow{BG}|^2+|\overrightarrow{CG}|^2)$ is equal to