For any vector $$\overrightarrow p ,$$ the value of $$\frac{3}{2}\left\{ {{{\left| {\overrightarrow p \times \hat i} \right|}^2} + {{\left| {\overrightarrow p \times \hat j} \right|}^2} + {{\left| {\overrightarrow p \times \hat k} \right|}^2}} \right\}{\text{ is :}}$$

Solution :
$$\eqalign{ & {\text{Suppose, }}\overrightarrow p = {p_1}\hat i + {p_2}\hat j + {p_3}\hat k \cr & \overrightarrow p \times \hat i = {p_2}\hat j \times \hat i + {p_3}\hat k \times \hat i = - {p_2}\hat k + {p_3}\hat j \cr & {\left| {\overrightarrow p \times \hat i} \right|^2} = p_2^2 + p_3^2 \cr & {\text{Similarly, }}{\left| {\overrightarrow p \times \hat j} \right|^2} = p_3^2 + p_1^2 \cr & {\text{and }}{\left| {\overrightarrow p \times \hat k} \right|^2} = p_1^2 + p_2^2 \cr & \therefore \,\frac{3}{2}\left\{ {{{\left| {\overrightarrow p \times \hat i} \right|}^2} + {{\left| {\overrightarrow p \times \hat j} \right|}^2} + {{\left| {\overrightarrow p \times \hat k} \right|}^2}} \right\} \cr & = 3\left( {p_1^2 + p_2^2 + p_3^2} \right) \cr & = 3{\overrightarrow p ^2} \cr} $$