Question

Three non-zero, non-parallel coplanar vectors are always
A. Linearly dependent
B. Linearly Independent
C. Either A or B
D. Cannot be determined

Answer 1

Hint: Let us use the property of coplanarity of three vectors to find whether they are linearly dependent or not.

Complete step-by-step answer:

Let the three vectors that are non-zero, non-parallel coplanar vectors be \[\vec x\], \[\vec y\] and \[\vec z\].

As we know that according to the property of coplanarity.

If number of vectors are more than three then,

Vectors are coplanar if among them no more than two are linearly independent vectors.

If the number of vectors are three and are (\[\vec A\], \[\vec B\] and \[\vec C\])

Then the three vectors are coplanar if their scalar triple product i.e. \[\left( {\vec A \times \vec B} \right).\vec C\] is equal to zero.

And they can be written as \[\vec A = \lambda \vec B + \mu \vec C\], where \[\lambda\] and \[\mu\] are any real numbers.

And according to the definition of linearly dependent vectors, vectors are said to be linearly independent if and only if one of them can be written in the form of others and all of them are non-zero vectors.

So, as we are given that three vectors (\[\vec x\], \[\vec y\] and \[\vec z\]) are coplanar.

So, according to property of coplanarity of three vectors they can be written as \[\vec x = \lambda \vec y + \mu \vec z\]

So, \[\vec x\] is written in form of \[\vec y\] and \[\vec z\]

And according to the definition of linearly dependent vectors we can say that vectors \[\vec x\], \[\vec y\] and \[\vec z\] are linearly dependent.

Hence, the correct option will be A.

Note: Whenever we come up with this type of problem where we are given some vectors and asked to find whether they are dependent or not we have to use the property of coplanarity of three vectors to get the required answer. And this will be the easiest and efficient way to find the solution of the problem.