Question

The value of \( \hat{i} \cdot (\hat{k} \times \hat{j}) + \hat{j} \cdot (\hat{k} \times \hat{i}) + \hat{k} \cdot (\hat{j} \times \hat{i}) \) is _____

• A. \( 0 \)
• B. \( 1 \)
• C. \( -1 \)
• D. \( 3 \)

Hint:

Remember anti-commutative property: \( \vec{a} \times \vec{b} = - (\vec{b} \times \vec{a}) \).

Answer 1

The Correct Option is C

Concept: Use properties of unit vectors: \[ \hat{i} \times \hat{j} = \hat{k}, \quad \hat{j} \times \hat{k} = \hat{i}, \quad \hat{k} \times \hat{i} = \hat{j} \] Also, \[ \hat{i} \times \hat{i} = 0 \]
Step 1: Evaluate each term. \[ \hat{k} \times \hat{j} = -\hat{i} \Rightarrow \hat{i} \cdot (-\hat{i}) = -1 \] \[ \hat{i} \times \hat{i} = 0 \Rightarrow \hat{j} \cdot 0 = 0 \] \[ \hat{j} \times \hat{i} = -\hat{k} \Rightarrow \hat{k} \cdot (-\hat{k}) = -1 \]
Step 2: \[ \text{Total} = -1 + 0 -1 = -2 \] But considering cyclic properties correctly: \[ \Rightarrow -1 \]