On the linear dependence of three coplanar vectors

Question

The general consensus seems to be that any three coplanar vectors are linearly dependent. Here's one source that says so.

However, considering three vectors, of which two are collinear and the third non-collinear to either of them, isn't that false? I've tried it with $\hat i,2 \hat i$ and $ \hat i+ \hat j$. They seem to be linearly independent.

Is it something about the definition that makes this case not applicable to the general statement?

Answer 1

1. Definitions:

   • Vectors are coplanar iff they are parallel to some plane. So, in two dimensions, all vectors are coplanar.
   • A set of vectors is linearly dependent iff some non-trivial linear combination of some of its members equals the zero vector. ^{(A trivial linear combination means that the coefficients are all 0.)}

2. As shown by Chris in another answer, your three given coplanar vectors are indeed linearly dependent.

3. A consequence of the second definition above is that if a set of vectors contains a linearly dependent pair (i.e., a collinear pair), then the entire set must also be linearly dependent.

Answer 2

A plane is two-dimensional, so any three vectors in a plane are dependent.

In your example, they are dependent since $2\hat i=2\cdot\hat i$. In other words, we get the non-trivial linear combination $(-2)\cdot\hat i+(1)\cdot2\hat i+0\cdot(\hat i+\hat j)=0$.

Answer 3

I believe the statement says any three coplanar vectors that are linearly dependent are coplanar.