Proving That Vectors Are Coplanar
A plane is a two dimensional space and two vectors are sufficient to describe it, meaning that any point in th plane can be expressed as a sum of two vectors (relative to some fixed point). If therefore there are three vectors all of which may be moved to lie lie in the plane, one of them is redundant and can be discarded. The vectors lie in the same plane, so are said to be coplanar or linearly independent.
This means that one of the vectors can be expressed as a linear combination of the other two. Suppose the three vectors areandthenfor some a and b.
This can be easily related to properties of matrices. If a column in a matrix can be expressed as a linear combination of the other two rows, then the determinant of the matrix is zero. A similar statement can be made for the rows. Conversely, if the determinant of a matrix is zero then one of the columns can be expressed as a linear combination of the others.
Writingandas the columns of a matrix gives the matrixThe determinant of this matrix is (expanding along the top row)
The determinant of a matrix is the same as the determinant of the transpose of the matrix (where the columns are written as rows). The calculation is exactly the same if we write the columns of the above matrix as rows and expand along the first column.
In general the test for linear independence is to write the vectors as column vectors (or row vectors) and find the determinant of the corresponding matrix. This only works of course if the resulting matrix is square.
If the number of vectors is greater than the dimension of the vectors (the number of rows for a column vector), then the vectors are necessarily linearly dependent. If the number of vectors is less than the dimension of the vectors (the number of rows for a column vector), then the vectors may or may not be linearly dependent.