In a $\triangle A B C$, if $\frac{a}{\tan A}=\frac{b}{\tan B}=\frac{c}{\tan C}$, then $\cos ^2 A+\cos ^2 B+\cos ^2 C=$

In $\triangle A B C$, if $a=7, b=10$ and $c=11$, then $\frac{R}{r}=$

Let $A B C$ be a triangle and $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ be the position vectors of $A, B$ and $C$, respectively. Let $D$ divides $B C$ in the ratio $3: 1$ internally and $E$ divides $A D$ in the ratio $4: 1$ internally. Let $B E$ meet $A C$ in $F$. If $E$ divides $B F$ in the ratio $3: 2$ internally, then the position vector of $F$ is

If $\alpha, \beta$ and $\gamma$ are real numebrs such that

$$ \begin{aligned} & \left(\frac{7}{3}+\beta\right) \hat{\mathbf{i}}-\hat{\mathbf{j}}+(\alpha+\gamma) \hat{\mathbf{k}} \\ & =\frac{5}{3}(\alpha \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}})+\beta(2 \hat{\mathbf{j}}+\hat{\mathbf{k}})+(\hat{\mathbf{i}}+\gamma \hat{\mathbf{j}}+3 \hat{\mathbf{k}}), \text { then } \\ & 5 \alpha-9 \beta+13 \gamma= \end{aligned} $$