Why is this theorem on coplanar vectors true (LINEAR ALGEBRA)?

[BlueRope]

A necessary and sufficient condition for three vectors to be coplanar is the equality is that the determinant of the matrix equals zero.

 

[Matterwave]

Basically that just says that $A\cdot (B\times C)=0$ (or any reordering thereof). BXC is a vector that is perpendicular to B and C. If A is coplanar with B and C, then it can be expressed as a linear combination of the two, i.e. A=bB+cC where b and c are real numbers. In that case, then it's obvious that A dotted into this vector which is perpendicular to both B and C would be 0.

Another way to think about it is to note that the above triple product has a value which is the volumn of the parallelepiped defined by A, B and C. If A, B and C are coplanar, then the parallelepiped has 0 volume.

 

[Jim Kata]

What Matterwave said is correct it follows from a theorem that says the determinant of a matrix is non zero if and only if the vectors which make it up are all linearly independent. So if the vectors are coplanar obviously one the vectors is linearly dependent so the determinant of the matrix they form must be zero.

 

[Deveno]

do this, find this determinant:

$$\begin{vmatrix}x_1&y_1&ax_1+by_1\\x_2&y_2&ax_2+by_2\\x_3&y_3&ax_3+by_3 \end{vmatrix}$$

the results should be enlightening.

 

[blue_raver22]

The theorem on coplanar vectors is true because it is based on the fundamental principles of linear algebra. In linear algebra, we often deal with vectors in three-dimensional space, and it is important to determine whether these vectors lie in the same plane or not. This is where the concept of coplanarity comes into play.

The determinant of a matrix is a measure of its invertibility, and when it equals zero, it signifies that the matrix is singular and cannot be inverted. In the context of coplanar vectors, this means that the three vectors can be expressed as linear combinations of each other, and therefore, they lie in the same plane.

Conversely, if the determinant of the matrix formed by the three vectors is non-zero, then the vectors are linearly independent and cannot be expressed as linear combinations of each other. This implies that they do not lie in the same plane and are not coplanar.

Therefore, the equality of the determinant to zero is not only a necessary but also a sufficient condition for three vectors to be coplanar. This theorem is crucial in various applications of linear algebra, such as in solving systems of linear equations and in determining the geometric properties of vectors in three-dimensional space.