Question

The value of \( \vec{i} \cdot (\vec{j} \times \vec{k}) + \vec{j} \cdot (\vec{i} \times \vec{k}) + \vec{k} \cdot (\vec{i} \times \vec{j}) \) is:

• A. 1
• B. 3
• C. -1
• D. 0

Hint:

Follow the circle clockwise for positive results and anti-clockwise for negative results.

Answer 1

The Correct Option is A

Step 1: Concept
Use the cyclic properties of unit vectors: $\vec{i} \times \vec{j} = \vec{k}$, $\vec{j} \times \vec{k} = \vec{i}$, $\vec{k} \times \vec{i} = \vec{j}$. Also, $\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a})$.
Step 2: Analysis
* $\vec{i} \cdot (\vec{j} \times \vec{k}) = \vec{i} \cdot \vec{i} = 1$. * $\vec{j} \cdot (\vec{i} \times \vec{k}) = \vec{j} \cdot (-\vec{j}) = -1$. * $\vec{k} \cdot (\vec{i} \times \vec{j}) = \vec{k} \cdot \vec{k} = 1$.
Step 3: Conclusion
Sum $= 1 - 1 + 1 = 1$.
Final Answer: (A)