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 \(Q\).
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 \(Q\).
Show Hint
Use vector identities involving dot and cross products and unit vectors to relate sums of squares of components.
Updated On: Jun 6, 2025
- \(P = Q\)
- \(P = 2Q\)
- \(P = 3Q\)
- \(P = 4Q\)
Show Solution
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The Correct Option is B
Solution and Explanation
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, \(P = 2Q\).
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