1. Explain what it means for three vectors in 3-space to be coplanar.?

Question

1. Explain what it means for three vectors in 3-space to be coplanar.

2. Describe how you can determine if three given vectors in 3-space are coplanar.

Answer 1

It means that there is a two-dimensional subspace of R³ that contains all of them. It is equivalent to the condition that they be linearly dependent (n.b. this is only because there are only three vectors. Any four vectors in R³ are linearly dependent, but are not necessarily coplanar).

The basic method of determining coplanarity is to find the determinant of the 3×3 matrix whose columns are the three vectors being tested. The vectors are coplanar iff the determinant is zero.

Edit: Corrected an error. The vectors are coplanar if the determinant IS zero, not the other way around. Ignore the previous version.

Answer 2

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Answer 3

3 vectors r coplanar means that they all lie in the same plane...i.e. either xy,yz or zx-plane

condition:if a,b,c are 3 vectors
[a b c]=0...i.e. a.(b cross c)=0