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

The value of $$\frac{(\mathbf{a} \times \mathbf{b})^2+(\mathbf{a} \cdot \mathbf{b})^2}{2(\mathbf{a})^2(\mathbf{b})^2}$$ is

Let $$\mathbf{a}=\hat{\mathbf{i}}-\hat{\mathbf{j}}, \mathbf{b}=\hat{\mathbf{j}}-\hat{\mathbf{k}}$$ and $$\mathbf{c}=\hat{\mathbf{k}}-\hat{\mathbf{i}}$$ if $$\mathbf{d}$$ is a unit vector such $$\mathbf{a} \cdot \mathbf{b}=0=[\mathbf{b} \mathbf{c} \mathbf{d}]$$, then $$\mathbf{d}$$ is

Let $$u$$ and $$v$$ be two non-zero vectors in $$R^3$$ with the intermediate angle $$45^{\circ}$$. Then $$|\mathbf{u} \times \mathbf{v}|$$ is equal to