In the triangle $ABC,$ if $\overrightarrow{AB} = a, \overrightarrow{AC} = c, \overrightarrow{BC} = b$,then which of the following is correct?

The vectors $\overrightarrow{AB} = 3 \hat{i} + 4 \hat{k}$ and $\overrightarrow{AC} = 5 \hat{i} - 2 \hat{j} + 4 \hat{k}$ are the sides of a triangle $ABC$. The length of the median through $A$ is

Let $\overrightarrow{OA} = \hat{i} - 3\hat{j} + \hat{k}$,$\overrightarrow{OB} = \hat{i} + 3\hat{j} - 2\hat{k}$,and $\overrightarrow{OC} = 4\hat{i} + 3\hat{j} + 5\hat{k}$ be the position vectors of three points $A$,$B$,and $C$. Let $P$ be the point which divides $AB$ in the ratio $2:1$. If $l, m, n$ are the direction cosines of the vector $\overrightarrow{PC}$,then $l + 3m + 2n =$

If $C$ is the mid-point of the line segment $AB$ and $P$ is any point outside the line $AB$,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

The direction ratios of the line bisecting the angle between the $x$-axis and the line having direction ratios $(3, -1, 5)$ are