For any vector p , the value of [2 | p |^2+| p |^2+| p k |^2 ] is

Step-by-step solution

Let p = x i + y j + z k . Then | p |^2 = x^2 + y^2 + z^2 . Calculate the cross products: p i = (x i + y j + z k ) i = -y k + z j . Therefore, | p i |^2 = (-y)^2 + z^2 = y^2 + z^2 . Similarly: p j = (x i + y j + z k ) j = x k - z i . Therefore, | p j |^2 = x^2 + (-z)^2 = x^2 + z^2 . And: p k = (x i + y j + z k ) k = -x j + y i . Therefore, | p k |^2 = y^2 + x^2 = x^2 + y^2 . Summing these squares: | p i |^2 + | p j |^2 + | p k |^2 = (y^2 + z^2) + (x^2 + z^2) + (x^2 + y^2) = 2(x^2 + y^2 + z^2) = 2| p |^2 . The expres