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
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
Show Hint
For unit vectors, their dot products follow specific relationships. Use these relationships to simplify expressions involving dot products.
Updated On: Mar 25, 2026
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The Correct Option is C
Solution and Explanation
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: \[ \boxed{\frac{3}{2}} \]
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