Question

If $a$ and $b$ are two non-zero non-collinear vectors then $a+3b$ and $a-3b$ are:

• A. Linearly independent.
• B. Linearly dependent.
• C. May be both
• D. None of these

Answer 1

The correct option is A Linearly independent.

Assume that $a+3b$ and $a-3b$ are linearly dependent

So we get $\alpha (a+3b)+\beta (a-3b)=0$

$\Rightarrow a(\alpha +\beta )+b(3\alpha -3\beta )=0$

Let us take $\frac{3\beta -3\alpha }{\alpha +\beta }=k$

So we get $a=kb$

But given $a$ and $b$ are non collinear , so our assumption is wrong

Therefore given vectors are linearly independent

So the correct option is $A$