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Question

Which of the following statements is / are true?
1.Set of two vectors a,b is linearly dependent if and only if either any of a and b is zero or they are parallel
2.a,b and c are linearly dependent a,b and c are coplanar.

A
Both 1 and 2 are false
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B
1 is true and 2 is false
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C
Both 1 and 2 are true
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D
2 is true and 1 is false
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Solution

The correct option is C Both 1 and 2 are true
Two vectors a and b are linearly dependent if for x,y ϵ R: xa+yb=0 where x and y are not simultaneously zero.
For linear dependence following cases are possible
x=0,y0 b=0
x0,y=0 a=0
x and y 0 The case where the vectors are parallel.
So xa+yb=0 if either a and b and parallel or one of them is zero.
Statement 2:
In the statement it is given that
a,b and c are linearly dependent. That means one of them can be written as a linear combination of other two.
So xa+yb=c
From fundamental theorem in 2D, we know that any linear combination of two vectors, gives a vector the same plane as that of a and b.
xa+yb is coplanar with a and b
Let's say xa+yb=c
So a, b and c are coplanar


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