Lagrange's Identity

If $\overrightarrow{a}$ is a unit vector such that $\overrightarrow{a}\times \hat{i}=\hat{j}$, find $\overrightarrow{a}.\hat{i}$.

If $\left|\overrightarrow{a}\right|=13,\left|\overrightarrow{b}\right|=5$ and $\overrightarrow{a}·\overrightarrow{b}=60,$ then find $|\overrightarrow{a}\times \overrightarrow{b}|$

If $\left|\overrightarrow{a}\right|=2,\left|\overrightarrow{b}\right|=5$ and $|\overrightarrow{a}\times \overrightarrow{b}|=8,$ find $\overrightarrow{a}·\overrightarrow{b}$

If $|\overrightarrow{a}\times \overrightarrow{b}|=4,|\overrightarrow{a}·\overrightarrow{b}|=2$, then $\left|\overrightarrow{a}{|}^2\right|\overrightarrow{b}{|}^2=$

If $\left|\overrightarrow{a}\right|=10,\left|\overrightarrow{b}\right|=2$ and $|\overrightarrow{a}\times \overrightarrow{b}|=16,$ find $\overrightarrow{a}·\overrightarrow{b}$

If $\overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k},$ then write the value of ${\left|\overrightarrow{r}\times \hat{i}\right|}^2.$

For any two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ write the value of ${\left(\overrightarrow{a}.\overrightarrow{b}\right)}^2+{\left|\overrightarrow{a}\times \overrightarrow{b}\right|}^2$ in terms of their magnitudes.

For any two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, prove that ${\left|\overrightarrow{a}\times \overrightarrow{b}\right|}^2=\left|\begin{array}{cc}\overrightarrow{a}.\overrightarrow{a}& \overrightarrow{a}.\overrightarrow{b}\\ \overrightarrow{b}.\overrightarrow{a}& \overrightarrow{b}.\overrightarrow{b}\end{array}\right|$