If in $$\triangle A B C, a \tan A+b \tan B=(a+b). \tan \left(\frac{A+B}{2}\right)$$, then which of the following holds?

If $$\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$$ and $$\mathbf{c}=x \hat{\mathbf{i}}+(x-2) \hat{\mathbf{j}}-\hat{\mathbf{k}}$$ and if the vector $$\mathbf{c}$$ lies in the plane of vectors $$\mathbf{a}$$ and $$\mathbf{b}$$ and then $$x$$ equals

Let $$u=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}$$ and $$v=3 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}$$. Consider three points $$P, Q$$ and $$R$$ having the position vectors $$\left(\frac{5}{2}\right) \hat{\mathbf{i}}-2 \hat{\mathbf{j}} ;\left(\frac{7}{3}\right) \hat{\mathbf{i}}-\hat{\mathbf{j}}$$ and $$\left(\frac{9}{4}\right) \hat{\mathbf{i}}$$ respectively. Among these, the points in the line passing through $$u$$ and $$v$$ are

The point of intersection of the lines joining points $$\hat{\mathbf{i}}+2 \hat{\mathbf{j}}, 2 \hat{\mathbf{i}}-\hat{\mathbf{j}}$$ and $$-\hat{\mathbf{i}}, 2 \hat{\mathbf{i}}$$ is