The perimeter of a triangle with sides represented by vectors $\vec{a} = 3\hat{i} + 4\hat{j} + 5\hat{k}$,$\vec{b} = 4\hat{i} - 3\hat{j} - 5\hat{k}$,and $\vec{c} = 7\hat{i} + \hat{j}$ is:

For any two points $M$ and $N$ in the $XY$-plane, let $\overrightarrow{MN}$ denote the vector from $M$ to $N$, and $\overrightarrow{0}$ denote the zero vector. Let $P, Q$ and $R$ be three distinct points in the $XY$-plane. Let $S$ be a point inside the triangle $\triangle PQR$ such that $\overrightarrow{SP} + 5\overrightarrow{SQ} + 6\overrightarrow{SR} = \overrightarrow{0}$. Let $E$ and $F$ be the mid-points of the sides $PR$ and $QR$, respectively. Then the value of $\frac{\text{length of the line segment } EF}{\text{length of the line segment } ES}$ is: (in $.20$)

The unit vector in the direction of $\vec{x} = (2, 3, \sqrt{3})$ is . . . . . . .

If $r \cdot i = r \cdot j = r \cdot k$ and $|r| = 3$, then $r = $

Let $A, B, C$ be distinct points with position vectors $\hat{i} + \hat{j}$,$\hat{i} - \hat{j}$,and $p\hat{i} - q\hat{j} + r\hat{k}$ respectively. If points $A, B, C$ are collinear,then which of the following can be correct?

If the vectors $\vec{a} = \lambda \hat{i} + 2\hat{j} - 3\hat{k}$ and $\vec{b} = \sqrt{\lambda} \hat{i} + \sqrt{13} \hat{j}$ have the same magnitude,then the value of $\lambda$ is: