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

Hint: The vectors parallel to the same plane or lie on the same plane are called coplanar vectors. The scalar triple product of a system of vectors will be zero only if they lie on the same plane.

Complete step by step answer:

The coplanar vectors are the vectors the lie on the same plane or are parallel to the same plane. Given, a system of vectors is said to be coplanar if their scalar triple product is zero. The scalar triple product can be written as $\left[ {\overrightarrow {{\text{a}}{\text{.}}} \overrightarrow {\text{b}} .\overrightarrow {\text{c}} } \right]$ or$\overrightarrow {{\text{a}}{\text{.}}} \left( {\overrightarrow {\text{b}} \times \overrightarrow {\text{c}} } \right)$ .Here, cross product of two vectors happens so a general vector (let us say $\overrightarrow {\text{d}} $ ) is generated and that vector has dot product with the third vector. The general vector generated ($\overrightarrow {\text{d}} $) will be perpendicular to the third vector $\overrightarrow {\text{a}} $ as the cross product of 2 vectors$\left( {\overrightarrow {\text{b}} \times \overrightarrow {\text{c}} } \right)$ give a perpendicular vector. The dot product of two vectors is zero if they are perpendicular to each other which means that $\overrightarrow {{\text{a}}{\text{.}}} \overrightarrow {\text{d}} = 0$ .So I statement is true.
Now given, a system of vectors is said to be coplanar if they are linearly dependent. If the vectors lie on the same plane then we can easily find ${\text{a,b,c}}$ and if two vectors are not parallel then the third vector can be expressed in the terms of the other two vectors. Therefore, they are linearly dependent. So II statement is also correct.
Hence the correct answer is ‘C’.

Note: The conditions for vectors to be coplanar if there are 3 vectors, is- a) if their scalar triple product is zero, b) if they are linearly dependent and c) In case of n vectors if no more than two vectors are linearly independent.