Question

A system of vectors is said to be coplanar, if
I. Their scalar triple product is zero.
II. They are linearly dependent.
Which of the following is true?

• A. Only I
• B. Only II
• C. Both I and II
• D. None of these

Answer 1

The correct option is C Both I and II

A system of vectors is said to be co-planar if their scalar triple product is zero because in scalar triple product cross product of two vectors happens and that vector is done dot product with the third vector. If the cross product of first two vectors happens, then a normal vector is generated and that vector if has a dot product with the third vector will be zero, if it lies in the same plane with the first two vectors since it will be perpendicular.

So, I is correct.

A system of vectors is said to be co-planar if they are linearly dependent because, in $1^{st}$ case, if all the three vectors lie in same line, then we can find appropriate $a,b,c$, since they lie in same plane and will be linearly dependent

In $2^{nd}$ case two vectors are not parallel. So, the third vector lying in the same plane can be expressed in terms of those two vectors. Thus they are linearly dependent and are co-planar.

So, II is correct.