Question

For any vector \( \vec{p} \), the value of \( 2\left[|\vec{p}\times \hat{i}|^2 + |\vec{p}\times \hat{j}|^2 + |\vec{p}\times \hat{k}|^2\right] \) is

• A. \( 4|\vec{p}|^2 \)
• B. \( 2|\vec{p}|^2 \)
• C. \( 4|\vec{p}| \)
• D. \( 2|\vec{p}| \)

Hint:

Use \( |\vec{a}\times\vec{b}|^2 = |\vec{a}|^2|\vec{b}|^2 - (\vec{a}\cdot\vec{b})^2 \) or component method for unit vectors.

Answer 1

The Correct Option is A

Step 1: Let the vector be.
\[ \vec{p} = ai + bj + ck \]

Step 2: Compute \( \vec{p} \times \hat{i} \).
\[ \vec{p} \times \hat{i} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} a & b & c 1 & 0 & 0 \end{vmatrix} \]
\[ = (0 - 0)\hat{i} - (0 - c)\hat{j} + (0 - b)\hat{k} \]
\[ = c\hat{j} - b\hat{k} \]
\[ |\vec{p}\times \hat{i}|^2 = b^2 + c^2 \]

Step 3: Compute \( \vec{p} \times \hat{j} \).
\[ \vec{p} \times \hat{j} = -c\hat{i} + a\hat{k} \]
\[ |\vec{p}\times \hat{j}|^2 = c^2 + a^2 \]

Step 4: Compute \( \vec{p} \times \hat{k} \).
\[ \vec{p} \times \hat{k} = b\hat{i} - a\hat{j} \]
\[ |\vec{p}\times \hat{k}|^2 = a^2 + b^2 \]

Step 5: Add all three.
\[ (b^2 + c^2) + (c^2 + a^2) + (a^2 + b^2) \]
\[ = 2(a^2 + b^2 + c^2) \]

Step 6: Multiply by 2.
\[ 2 \times 2(a^2 + b^2 + c^2) = 4(a^2 + b^2 + c^2) \]

Step 7: Final conclusion.
\[ |\vec{p}|^2 = a^2 + b^2 + c^2 \]
\[ \boxed{4|\vec{p}|^2} \]