Question

Given the following expression
A) \((\overset{⃗}{a}\times \overset{⃗}{b})\cdot \overset{⃗}{c}\)
B) \(\overset{⃗}{a}\times (\overset{⃗}{b}\cdot \overset{⃗}{c})\)
C) \(\overset{⃗}{a}\cdot (\overset{⃗}{b}\cdot \overset{⃗}{c})\)
D) \(|\overset{⃗}{a}|(\overset{⃗}{b}\cdot \overset{⃗}{c})\)
E) \((\overset{⃗}{a}\cdot \overset{⃗}{b})\times (\overset{⃗}{b}\cdot \overset{⃗}{c})\)
Then which of the following is not correct

• A. B and E are meaningful
• B. A and D are meaningful
• C. B, C and E are meaningless
• D. A is meaningful but B is meaningless

Hint:

A cross product needs two vectors and a dot product gives a scalar; check that each operation is defined.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
A dot product of two vectors is a scalar. A cross product is defined only between two vectors. So a cross product of a vector with a scalar, or of two scalars, has no meaning.

Step 2: Check each expression
A) \((\vec a\times\vec b)\cdot\vec c\): cross first gives a vector, then dot with \(\vec c\) gives a scalar. Meaningful.
B) \(\vec a\times(\vec b\cdot\vec c)\): \(\vec b\cdot\vec c\) is a scalar, and a vector cannot be crossed with a scalar. Meaningless.
C) \(\vec a\cdot(\vec b\cdot\vec c)\): a vector dotted with a scalar. Meaningless.
D) \(|\vec a|(\vec b\cdot\vec c)\): scalar times scalar. Meaningful.
E) \((\vec a\cdot\vec b)\times(\vec b\cdot\vec c)\): cross product of two scalars. Meaningless.

Step 3: Compare with the options
The statement "B and E are meaningful" is not correct, because both are meaningless. That is option (A).

Final Answer:
Expressions B and E are meaningless, so the claim that they are meaningful is the incorrect one, option (A). \[ \boxed{\text{(A)}} \]