If \( \vec{a} \), \( \vec{b} \) and \( \vec{c} \) are three vectors, then meaningless expressions are
(A) \( \vec{a}.(\vec{b}\times\vec{c}) \)
(B) \( \vec{a}.(\vec{b}.\vec{c}) \)
(C) \( \vec{a}\times(\vec{b}\times\vec{c}) \)
(D) \( \vec{a}\times(\vec{b}.\vec{c}) \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (B) and (D) only
The question asks for the meaningless expressions. The dot product and the cross product are defined only between two vectors; a scalar can only multiply a vector.
So \( \vec{b}\times\vec{c} \) is a vector, while \( \vec{b}\cdot\vec{c} \) is a scalar.
Checking the statements
(A) \( \vec{a}\cdot(\vec{b}\times\vec{c}) \) — Correct. A vector dotted with the vector \( \vec{b}\times\vec{c} \): the scalar triple product, which is meaningful.
(B) \( \vec{a}\cdot(\vec{b}\cdot\vec{c}) \) — Incorrect. \( \vec{b}\cdot\vec{c} \) is a scalar, and a dot product of a vector with a scalar is not defined, so it is meaningless.
(C) \( \vec{a}\times(\vec{b}\times\vec{c}) \) — Correct. A vector crossed with the vector \( \vec{b}\times\vec{c} \): the vector triple product, which is meaningful.
(D) \( \vec{a}\times(\vec{b}\cdot\vec{c}) \) — Incorrect. A cross product of a vector with the scalar \( \vec{b}\cdot\vec{c} \) is not defined, so it is meaningless.
The meaningless expressions are (B) and (D).