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

Step 1: Understanding the Concept:
We use the cyclic properties of unit vectors: $\hat{i} \times \hat{j} = \hat{k}$, $\hat{j} \times \hat{k} = \hat{i}$, and $\hat{k} \times \hat{i} = \hat{j}$. If the order is reversed, the sign is negative.
Step 2: Formula Derivation:
1. $\hat{k} \times \hat{j} = -\hat{i}$
2. $\hat{k} \times \hat{i} = \hat{j}$
3. $\hat{j} \times \hat{i} = -\hat{k}$
Step 3: Explanation:
Substitute these back: $\hat{i} \cdot (-\hat{i}) + \hat{j} \cdot (\hat{j}) + \hat{k} \cdot (-\hat{k})$ $(-1) + (1) + (-1) = -1$
Step 4: Final Answer:
The value is $-1$.