Question

If \( \vec{a} = \lambda\hat{i} + 2\hat{j} + 2\hat{k} \) and \( \vec{b} = 2\hat{i} + 2\hat{j} + \lambda\hat{k} \) are at right angle, then the value of \( |\vec{a}+\vec{b}| - |\vec{a}-\vec{b}| \) is

• A. 2
• B. 1
• C. 0
• D. -1
• E. -2

Hint:

If vectors are perpendicular, magnitudes of sum and difference are equal.

Answer 1

The Correct Option is C

Concept:
• If vectors are perpendicular: \[ \vec{a}\cdot\vec{b} = 0 \]
• Identity: \[ |\vec{a}+\vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 \] \[ |\vec{a}-\vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 \]

Step 1: Apply perpendicular condition.
\[ \vec{a}\cdot\vec{b} = 0 \] \[ \lambda(2) + 2(2) + 2(\lambda) = 0 \] \[ 2\lambda + 4 + 2\lambda = 0 \] \[ 4\lambda + 4 = 0 \Rightarrow \lambda = -1 \]

Step 2: Substitute vectors.
\[ \vec{a} = (-1,2,2), \quad \vec{b} = (2,2,-1) \]

Step 3: Compute \( \vec{a}+\vec{b} \).
\[ (1,4,1) \] \[ |\vec{a}+\vec{b}| = \sqrt{1+16+1} = \sqrt{18} \]

Step 4: Compute \( \vec{a}-\vec{b} \).
\[ (-3,0,3) \] \[ |\vec{a}-\vec{b}| = \sqrt{9+0+9} = \sqrt{18} \]

Step 5: Subtract.
\[ \sqrt{18} - \sqrt{18} = 0 \]

Step 6: Final Answer.
\[ \boxed{0} \]