Question

Let \(\vec a=i+j+k\), \(\vec b=i-j+k\). If \(\vec c \perp \vec a\), \(\vec d \parallel \vec a\), and \(\vec b=\vec c+\vec d\), then \((\vec c\times \vec d)^2=\)

• A. \(\frac{2}{9}\)
• B. 8
• C. \(\frac{4}{3}\)
• D. \(\frac{2}{3}\) \bigskip

Hint:

Perpendicular + parallel decomposition is key in such vector splits.

Answer 1

The Correct Option is C

Concept: Resolve vector into perpendicular and parallel components.

Step 1: Decompose vector.
\[ \vec d=\text{projection of }\vec b \text{ on } \vec a \] \[ \vec c=\vec b-\vec d \]

Step 2: Compute magnitudes.
\[ \vec a\cdot \vec b=1 \] \[ |\vec a|^2=3 \] \[ \vec d=\frac{1}{3}\vec a \]

Step 3: Compute cross product.
\[ |\vec c\times \vec d|=|\vec c||\vec d| \] \[ (\vec c\times \vec d)^2=\frac{4}{3} \] \[ \boxed{(C)} \]