If $C$ is the mid-point of line segment $AB$ and $P$ is any point not on the line $AB$,then

If $\bar{x}$ is a non-zero vector and $k < 0, k \neq -1$,then $\frac{k \bar{x}}{|\bar{x}|}$ is $.........$

If $\overline{a}=2 \hat{\imath}+3 \hat{\jmath}+\hat{k}$,$\overline{b}=4 \hat{\imath}+5 \hat{\jmath}+3 \hat{k}$ and $\overline{c}=6 \hat{\imath}+\hat{\jmath}+5 \hat{k}$ are the position vectors of the vertices of a triangle $ABC$ respectively,then the position vector of the intersection of the medians (centroid) of the triangle $ABC$ 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 $\vec{a} + 5\vec{b} = \vec{c}$ and $\vec{a} - 7\vec{b} = 2\vec{c}$,then which of the following is true?

If $\vec{a} = \hat{i} + \hat{j} - 2 \hat{k}$,$\vec{b} = 2 \hat{i} - \hat{j} + \hat{k}$ and $\vec{c} = 3 \hat{i} - \hat{k}$. If $\vec{c} = m \vec{a} + n \vec{b}$ then $m + n = $