Given any vector $\vec r,\,|\vec r×\hat i|^2+|\vec r×\hat j|^2+|\vec r×\hat k|^2$ equals:
Answer & explanation
Correct answer: option 3
$s=|\vec r×\hat i|^2+|\vec r×\hat j|^2+|\vec r×\hat k|^2$
$|\vec r×\hat i|^2=(\vec r×\hat i).(\vec r×\hat i)=(\vec r.\vec r)(\hat i.\hat i)-(\vec r.\hat i)(\hat i.\vec r)=|\vec r|^2-(r_x).(r_x)$
Similarly, find for others.
$s=3|\vec r|^2-(r_x^2+r_y^2+r_z^2)=2|\vec r|^2$