If two vertices of a triangle are $\hat{i} - \hat{j}$ and $\hat{j} + \hat{k}$,then the third vertex can be

The diagonals of a parallelogram are the vectors $\vec{d_1} = 3 \hat{i} + 6 \hat{j} - 2 \hat{k}$ and $\vec{d_2} = -\hat{i} - 2 \hat{j} - 8 \hat{k}$. Then the length of the shorter side of the parallelogram is

If $2 \hat{i}+4 \hat{j}-5 \hat{k}$, $\hat{i}+\hat{j}+\hat{k}$, and $\hat{j}+2 \hat{k}$ are the position vectors of the vertices $A$, $B$, and $C$ of a triangle respectively, then a unit vector along the median drawn through the vertex $A$ is

Given the following expressions: $(A)$ $(a \times b) \cdot c$, $(B)$ $a \times (b \cdot c)$, $(C)$ $a \cdot (b \cdot c)$, $(D)$ $|a|(b \cdot c)$, $(E)$ $(a \cdot b) \times (b \cdot c)$. Which of the following statements is correct regarding their mathematical meaning?

If non-zero vectors $\vec{a}$,$\vec{b}$,and $\vec{c}$ are related by $\vec{a} = 8\vec{b}$ and $\vec{c} = -7\vec{b}$,find the angle between $\vec{a}$ and $\vec{c}$.

Consider two points $P$ and $Q$ with position vectors $\overrightarrow{OP} = 3\vec{a} - 2\vec{b}$ and $\overrightarrow{OQ} = \vec{a} + \vec{b}$. Find the position vector of a point $R$ which divides the line joining $P$ and $Q$ in the ratio $2:1$ externally.