Question

If a and b be parallel vectors, then [a c b]
A) \[0\]
B) \[1\]
C) \[2\]
D) None of these

Answer 1

Hint: In this question we have to find the scalar triple product of three parallel vectors. Parallel vectors are those vectors which are having the same direction or opposite direction. Parallel vectors are also known as collinear vectors.
In parallel vectors, one vector is a multiple of some other vector.



Formula Used:\(\vec a = \lambda \vec b\)
\(\lambda \)is a scalar quantity.



Complete step by step solution:Given: vector a, b are the parallel vectors.
\(\vec a = \lambda \vec b\)
\(\lambda \)is a scalar quantity.
We know that:
\(\left[ {\vec a\vec b\vec c} \right] = \vec a{\rm{.}}\left( {\vec b \times \vec c} \right)\)
\(\left[ {\vec a\vec b\vec c} \right] = \vec a{\rm{.}}\left( {\vec b \times \vec c} \right)\)
\(\vec a{\rm{.}}\left( {\vec b \times \vec c} \right) = \lambda \vec b\left( {\vec b \times \vec c} \right)\)
\(\lambda \vec b\left( {\vec b \times \vec c} \right) = \left[ {\lambda \vec b\vec b\vec c} \right]\)
\(\left[ {\lambda \vec b\vec b\vec c} \right] = 0\)
 If two vectors in a scalar triple product are the same then the scalar triple product will be zero. \(\left[ {\lambda \vec b\vec b\vec c} \right] = \lambda \left[ {\vec b\vec b\vec c} \right] = \lambda \times 0 = 0\)
\(\left[ {\vec a\vec b\vec c} \right] = \left[ {\vec a\vec c\vec b} \right] = 0\)



Option ‘A’ is correct

Note: Result is a property of parallel vectors. If a question is asked to prove two vectors parallel among the given three collinear vectors then show that the scalar triple vector of three given vectors is zero.
In parallel vectors one vector is a scalar multiple of one of the given vectors.