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?

• A. $(\vec{a}+\vec{b})+\vec{c}=\vec{a}+(\vec{b}+\vec{c})$
• B. $m(\vec{a}+\vec{b}+\vec{c})=m\vec{a}+m\vec{b}+m\vec{c}$
• C. $(m+n)\vec{a}=m\vec{a}+n\vec{a}$
• D. $m(\vec{a} \cdot \vec{b}) = (m\vec{a}) \cdot (m\vec{b})$
• E. $m(\vec{a}\times\vec{b}) = (m\vec{a})\times\vec{b}$

Hint:

A scalar multiplier $m$ distributes only once across a dot or cross product.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We need to check the validity of five statements involving basic vector algebra properties, including vector addition, scalar multiplication, the dot product, and the cross product.
Step 2: Detailed Explanation:
Let's analyze each statement:
(A) \( (\vec{a}+\vec{b})+\vec{c = \vec{a}+(\vec{b}+\vec{c}) \)}
This is the associative law of vector addition. It is a fundamental property of vectors. This statement is true.
(B) \( m(\vec{a}+\vec{b}+\vec{c}) = m\vec{a+m\vec{b}+m\vec{c} \)}
This is the distributive law of scalar multiplication over vector addition. This statement is true.
(C) \( (m+n)\vec{a = m\vec{a}+n\vec{a} \)}
This is the distributive law of scalar addition over scalar multiplication with a vector. This statement is true.
(D) \( m(\vec{a} \cdot \vec{b}) = (m\vec{a}) \cdot (m\vec{b}) \)
Let's simplify the right-hand side (RHS). The dot product is linear, so we can factor out the scalars.
RHS: \( (m\vec{a}) \cdot (m\vec{b}) = m \cdot m (\vec{a} \cdot \vec{b}) = m^2 (\vec{a} \cdot \vec{b}) \).
The statement claims \( m(\vec{a} \cdot \vec{b}) = m^2(\vec{a} \cdot \vec{b}) \). This is only true if \( m = m^2 \) (i.e., \( m=0 \) or \( m=1 \)) or if \( \vec{a} \cdot \vec{b} = 0 \). It is not true for all scalars m and vectors \( \vec{a}, \vec{b} \). Therefore, this statement is not true.
(E) \( m(\vec{a} \times \vec{b}) = (m\vec{a}) \times \vec{b \)}
This is a property of scalar multiplication with the vector cross product. It is also equal to \( \vec{a} \times (m\vec{b}) \). This statement is true.
Step 3: Final Answer:
The statement that is not true is (D).