Linear Combination of Vectors - Practice Questions & MCQ

For example:

Linear Independent Vectors

It can be easily verified that

1. A pair of non-collinear vectors ( say a₁ and a₂ ) are linearly independent.

2. A triad of non-coplanar vector is linearly independent

If a, b, c are three non-zero, non-coplanar vectors and x, y, z are three scalars such that 

Proof: It is given that xa + yb + zc = 0     ......(i)

Suppose that  

Then Eq. (i) can be written as 

Now,  and   are scalars because x, y and are scalars. Thus, Eq. (ii) expresses a as a linear combination of b and c. Hence, a is coplanar with b and c which is contrary to our hypothesis because a, b and c are given to be non-coplanar. Thus, our supposition that is wrong.

Hence, x=0.

Similarly, we can prove that y=0 and z=0.

Note: 4 vectors are always linearly dependent

It can be easily verified that

1. A pair of collinear vectors is linearly dependent.

2. A triad of coplanar vectors is linearly dependent.

Test of collinearity of three points

Then the pointgs A, B and C are collinear

Proof:

Hence, the three point A, B and C are collinear. 

Theorem 1:

Proof: 

Theorem 2

NOTE:

1. 

(Proof of this will be seen in the concept of Scalar triple Product)

2.