Before we go on to solving examples involving the concepts we’ve seen till now, you are urged to once again go over the entire earlier discussion we’ve had, so that the “big picture” is clear in your mind.
Show that the vectors and are linearly independent.
Let be scalars such that
The determined of the coefficient matrix is
Thus, the system of equations in has no solution for and apart from the trivial solution This implies that the three vectors are linearly independent.
COLLINEARITY OF POINTS
Let and be three non-coplanar vectors. Prove that the points and are collinear. When we say the point , we mean the point whose position vector, i.e, the vector drawn from the origin to that point, is .
We have been given the position vectors of three points and we are required to prove that they are collinear. Let us see what condition must be satisfied in order for three points to be collinear:
Thus, there must be some for which
Since and are non-coplanar, we have
This consistently gives the solution , implying , and are collinear.