Question

If \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are three unit vectors such that \[ \mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0}, \quad \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{c} = \mathbf{c} \cdot \mathbf{a} \] then the value of \( \mathbf{a} \cdot \mathbf{a} \) is

• A. -3
• B. -2
• C. 3/2
• D. 0

Hint:

For unit vectors, their dot products follow specific relationships. Use these relationships to simplify expressions involving dot products.

Answer 1

The Correct Option is C

Step 1: Use the dot product properties.
From the given conditions, we know that \( \mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0} \), so \( \mathbf{a} + \mathbf{b} = -\mathbf{c} \). Taking the dot product of both sides with themselves and simplifying gives us \( \mathbf{a} \cdot \mathbf{a} = \frac{3}{2} \).
Step 2: Conclusion.
The value of \( \mathbf{a} \cdot \mathbf{a} \) is \( \frac{3}{2} \). Final Answer: \[ \boxed{\frac{3}{2}} \]