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

Given, $$\mathbf{a}=3 \hat{\mathbf{i}}-\hat{\mathbf{j}}, \mathbf{b}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-3 \hat{\mathbf{k}}$$ and $$\mathbf{b}=\mathbf{b}_1+\mathbf{b}_2$$ where $$\mathbf{b}_1$$ is parallel to $$\mathbf{a}$$ and $$\mathbf{b}_2$$ is perpendicular to $$\mathbf{a}$$. Then, $$\mathbf{b}_2$$ is equal to