Let $$a, b$$ and $$c$$ be unit vectors such that $$a$$ is perpendicular to the plane containing $$\mathbf{b}$$ and $$\mathbf{c}$$ and angle between $$\mathbf{b}$$ and $$\mathbf{c}$$ is $$\frac{\pi}{3}$$. Then, $$|\mathbf{a}+\mathbf{b}+\mathbf{c}|=$$

Let $$\mathbf{F}=2 \hat{i}+2 \hat{j}+5 \hat{k}, A=(1,2,5), B=(-1,-2,-3)$$ and $$\mathbf{B A} \times \mathbf{F}=4 \hat{i}+6 \hat{j}+2 \lambda \hat{k}$$, then $$\lambda=$$

$$O A B C$$ is a tetrahedron. If $$D, E$$ are the mid-points of $$O A$$ and $$B C$$ respectively, then $$\mathbf{D E}=$$

If $$\mathbf{a}+\mathbf{b}+\mathbf{c}=0$$ and $$|\mathbf{a}|=7,|\mathbf{b}|=5,|\mathbf{c}|=3$$ then the angle between $$\mathbf{b}$$ and $$\mathbf{c}$$ is