$\vec{a} \times \vec{b}=2 \hat{i}-\hat{k}$
$\vec{a} \cdot \vec{b}=3$
$|\vec{a} \times \vec{b}|^{2}+|\vec{a} \cdot \vec{b}|^{2}=|\vec{a}|^{2} \cdot|\vec{b}|^{2}$
$5+9=6|\vec{b}|^{2}$
$|b|^{2}=\frac{7}{3}$
$|\vec{a}-\vec{b}|=\sqrt{|\vec{a}|^{2}+|\vec{b}|^{2}-2 \vec{a} \cdot \vec{b}}=\sqrt{\frac{7}{3}}$
projection of $\vec{b}$ on $\vec{a}-\vec{b}=\frac{\vec{b} \cdot(\vec{a}-\vec{b})}{|\vec{a}-\vec{b}|}$
$=\frac{\vec{b} \cdot \vec{a}-|\vec{b}|^{2}}{|\vec{a}-\vec{b}|}=\frac{3-\frac{7}{3}}{\sqrt{\frac{7}{3}}}$
$=\frac{2}{\sqrt{21}}$
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$ \mathrm{L}_1: \overrightarrow{\mathrm{r}}=(2+\lambda) \hat{\mathrm{i}}+(1-3 \lambda) \hat{\mathrm{j}}+(3+4 \lambda) \hat{\mathrm{k}}, \lambda \in \mathbb{R} $
$ \mathrm{L}_2: \overrightarrow{\mathrm{r}}=2(1+\mu) \hat{\mathrm{i}}+3(1+\mu) \hat{\mathrm{j}}+(5+\mu) \hat{k}, \mu \in \mathbb{R}$
is $\frac{\mathrm{m}}{\sqrt{\mathrm{n}}}$, where $\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$, then the value of $\mathrm{m}+\mathrm{n}$ equals.
$\left( {{{\partial u} \over {\partial x}} + {{\partial u} \over {\partial y}} + {{\partial u} \over {\partial z}}} \right)$ $(x + y + z) =$
Which of the following properties does $R$ satisfy?
$I.$ Reflexivity $II.$ Symmetry $III.$ Transitivity