If $$\mathbf{a}+2 \mathbf{b}+3 \mathbf{c}=0$$ and $$(\mathbf{a} \times \mathbf{b})+(\mathbf{b} \times \mathbf{c})+(\mathbf{c} \times \mathbf{a})=\lambda(\mathbf{b} \times \mathbf{c})$$, then the value of $$\lambda$$ is equal to

If $$|\vec{a}+\vec{b}|=|\vec{a}-\vec{b}|$$, then

The component of $$\hat{\mathbf{i}}$$ in the direction of the vector $$\hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$$ is

If $$|\mathbf{a}|=2$$ and $$|\mathbf{b}|=3$$ and the angle between $$\mathbf{a}$$ and $$\mathbf{b}$$ is $$120^{\circ}$$, then the length of the vector $$\left|\frac{\mathbf{a}}{2}-\frac{\mathbf{b}}{3}\right|$$ is