Example – 6
Show that the vectors  are linearly independent.
Solution: Let l,m ,Î R be scalars such that
The determined of the coefficient matrix is
Thus, the system of equations in(1) has no solution for l,m and g apart from the trivial solution l = m = g = 0. This implies that the three vectors are linearly independent.
Example – 7 COLLINEARITY OF POINTS
Let  , and  be three non-coplanar vectors. Prove that the points A(2 + - ), B(5 - + 2)and
C(8 - 3 + 5 )
are collinear. When we say the point P(  ), we mean the point whose position vector, i.e, the vector drawn from the origin O to that point,  ,
is .
Solution: 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 l Î R for which
Since , and are non-coplanar, we have
3 - 3l = 0
2l - 2 = 0
3 - 3l = 0
This consistently gives the solution l =1, implying A, B and C are collinear.
Example – 8
Let , and be three non-coplanar vectors. Prove that the points A(2 + 3 - ), B( - 2 -3 ), C(3 + 4 - 2 ) and D( - 6 + 6 )
are coplanar.
Solution: As in the previous example, we first draw a visual picture to determine when four points can be coplanar.
Thus, as explained in the figure, we must have some scalars l,m Î R for which
Since , and are non-coplanar, we must have
1+l + m = 0
1+ 5l + 9m = 0
1+ 4l + 7m = 0
As can be easily verified, this system has the solution l = -2, m =1, implying  ,  and  are
indeed coplanar.
Thus, the points A, B, C and D are coplanar.
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