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

by BlueRope
Tags: algebra, coplanar, linear, theorem, vectors
 Sci Advisor P: 2,851 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.
 Sci Advisor P: 906 Why is this theorem on coplanar vectors true (LINEAR ALGEBRA)? 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.