Let $\mathbf{a}, \mathbf{b}, \mathbf{c}$ be three vectors such that $\mathbf{a} \cdot \mathbf{a}=\mathbf{b} \cdot \mathbf{b}=\mathbf{c} \cdot \mathbf{c}=5$ and $|\mathbf{a}+\mathbf{b}-\mathbf{c}|^2+|\mathbf{b}+\mathbf{c}-\mathbf{a}|^2+|\mathbf{c}+\mathbf{a}-\mathbf{b}|^2=50$, then $\mathbf{a} \cdot \mathbf{b}+\mathbf{b} \cdot \mathbf{c}+\mathbf{c} \cdot \mathbf{a}=$

Let $\mathbf{c}$ be a vector coplanar with the unit vectors $\mathbf{a}, \mathbf{b}$ and let $\mathbf{d}$ be the unit vector perpendicular to $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$. If $[\mathbf{a} \mathbf{b} \mathbf{d}] \mathbf{c}-[\mathbf{a} \mathbf{b} \mathbf{c}] \mathbf{d}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and the angle between $\mathbf{a}$ and $\mathbf{b}$ is $30^{\circ}$, then $|\mathbf{c}|=$

If $|\mathbf{a}|=4,|\mathbf{b}|=5$ and $|\mathbf{a}-\mathbf{b}|=3$ and $\theta$ is the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$, then $\cot ^2 \theta=$

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