Question:
Let $\vec{a}, \vec{b}, \vec{c}$ be any three vectors and $m, n$ be scalars. Which one of the following is not true?
Let $\vec{a}, \vec{b}, \vec{c}$ be any three vectors and $m, n$ be scalars. Which one of the following is not true?
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A scalar multiplier $m$ distributes only once across a dot or cross product.
Updated On: Apr 28, 2026
- $(\vec{a}+\vec{b})+\vec{c}=\vec{a}+(\vec{b}+\vec{c})$
- $m(\vec{a}+\vec{b}+\vec{c})=m\vec{a}+m\vec{b}+m\vec{c}$
- $(m+n)\vec{a}=m\vec{a}+n\vec{a}$
- $m(\vec{a} \cdot \vec{b}) = (m\vec{a}) \cdot (m\vec{b})$
- $m(\vec{a}\times\vec{b}) = (m\vec{a})\times\vec{b}$
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The Correct Option is D
Solution and Explanation
Step 1: Concept
The property for scalar multiplication with a dot product is $m(\vec{a} \cdot \vec{b}) = (m\vec{a}) \cdot \vec{b} = \vec{a} \cdot (m\vec{b})$.
Step 2: Analysis
In option (D), the RHS is $(m\vec{a}) \cdot (m\vec{b}) = m^2(\vec{a} \cdot \vec{b})$.
Step 3: Conclusion
Since $m(\vec{a} \cdot \vec{b}) \neq m^2(\vec{a} \cdot \vec{b})$ (unless $m=0, 1$), option (D) is incorrect. Final Answer: (D)
The property for scalar multiplication with a dot product is $m(\vec{a} \cdot \vec{b}) = (m\vec{a}) \cdot \vec{b} = \vec{a} \cdot (m\vec{b})$.
Step 2: Analysis
In option (D), the RHS is $(m\vec{a}) \cdot (m\vec{b}) = m^2(\vec{a} \cdot \vec{b})$.
Step 3: Conclusion
Since $m(\vec{a} \cdot \vec{b}) \neq m^2(\vec{a} \cdot \vec{b})$ (unless $m=0, 1$), option (D) is incorrect. Final Answer: (D)
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