The two vector $$\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$$ and $$\hat{\mathbf{i}}+3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$$ represent the two sides $$\overline{A B}$$ and $$\overline{A C}$$ respectively of a $$\triangle A B C$$. The length of the median through $$A$$ is

If $$\mathbf{a}$$ and $$\mathbf{b}$$ are unit vectors and $$\theta$$ is the angle between $$\mathbf{a}$$ and $$\mathbf{b}$$, then $$\sin \frac{\theta}{2}$$ is equal to

If $$|\mathbf{a}+\mathbf{b}|^2+|\mathbf{a} \cdot \mathbf{b}|^2=144|\mathbf{a}|=6$$, then $$|\mathbf{b}|$$ is equal to

If the vectors $$2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, 2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}$$ and $$\lambda \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$$ are coplanar, then the value of $$\lambda$$ is