Let $\mathbf{a} \times \mathbf{b}=7 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}-4 \hat{\mathbf{k}}$ and $\mathbf{a}=\hat{\mathbf{i}}+3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$. If the length of projection of $\mathbf{b}$ on $\mathbf{a}$ is $$ \frac{8}{\sqrt{14}}, \text { then }|b|= $$

Let $A B C$ be an equilateral triangle of side a. $M$ and $N$ are two points on the sides $A B$ and $A C$, respectively such that $\mathbf{A N}={ }^{\prime} K \mathbf{A C}$ and $\mathbf{A B}=3 \mathbf{A M}$. If the vectors $\mathbf{B N}$ and $\mathbf{C M}$ are perpendicular, then $K=$

Let $\mathbf{a}$ and $\mathbf{b}$ be two non-collinear vector of unit modulus. If $\mathbf{u}=\mathbf{a}-(\mathbf{a} \cdot \mathbf{b}) \mathbf{b}$ and $\mathbf{v}=\mathbf{a} \times \mathbf{b}$, then $|\mathbf{v}|=$

In a regular hexagon $A B C D E F, \mathbf{A B}=\mathbf{a}$ and $\mathbf{B C}=\mathbf{b}$, then $F A=$