Question

If

\[ P = (a \times \mathbf{i})^2 + (a \times \mathbf{j})^2 + (a \times \mathbf{k})^2 \]

and

\[ Q = (a \cdot \mathbf{i})^2 + (a \cdot \mathbf{j})^2 + (a \cdot \mathbf{k})^2, \]

Then find the relation between \(P\) and \(Q\).

• A. \(P = Q\)
• B. \(P = 2Q\)
• C. \(P = 3Q\)
• D. \(P = 4Q\)

Hint:

Use vector identities involving dot and cross products and unit vectors to relate sums of squares of components.

Answer 1

The Correct Option is B

Recall the identity: \[ |\mathbf{a} \times \mathbf{x}|^2 + (\mathbf{a} . \mathbf{x})^2 = |\mathbf{a}|^2 |\mathbf{x}|^2. \] For \(\mathbf{x} = \mathbf{i}, \mathbf{j}, \mathbf{k}\), each unit vector, \[ |\mathbf{x}|^2 = 1. \] Summing for \(\mathbf{i}, \mathbf{j}, \mathbf{k}\), \[ P + Q = 3 |\mathbf{a}|^2. \] But also, \[ Q = (a_x)^2 + (a_y)^2 + (a_z)^2 = |\mathbf{a}|^2, \] so \[ P + Q = 3Q \implies P = 2Q. \] Hence, \(P = 2Q\).