If the vectors $(2 \hat{\imath} - q \hat{\jmath} + 3 \hat{k})$ and $(4 \hat{\imath} - 5 \hat{\jmath} + 6 \hat{k})$ are collinear,then the value of $q$ is

If $\hat{i}+4 \hat{j}+3 \hat{k}$,$\hat{i}+2 \hat{j}+3 \hat{k}$,and $3 \hat{i}+2 \hat{j}+\hat{k}$ are position vectors of $A$,$B$,and $C$ respectively,and if $D$ and $E$ are midpoints of sides $BC$ and $AC$,then $\overrightarrow{DE}$ is equal to:

If $(\alpha, \beta, \gamma)$ is a triad of real numbers satisfying $\hat{i}-2 \hat{j}+5 \hat{k}=\alpha(\hat{i}+\hat{j}+\hat{k})+\beta(\hat{i}+2 \hat{j}+3 \hat{k})+\gamma(2 \hat{i}-\hat{j}+\hat{k}),$ then $\alpha^2-\beta^2+\gamma^2=$

Find the scalar components and magnitude of the vector joining the points $P(x_{1}, y_{1}, z_{1})$ and $Q(x_{2}, y_{2}, z_{2}).$

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 the orthocentre of the triangle whose vertices are $2 \hat{i}+3 \hat{j}+5 \hat{k}$,$5 \hat{i}+2 \hat{j}+3 \hat{k}$,and $3 \hat{i}+5 \hat{j}+2 \hat{k}$ is $x \hat{i}+y \hat{j}+z \hat{k}$,then: