Question

How do we find out if vectors are parallel?

Answer 1

Hint: In the given question, we have been asked to find out a way to know whether the given vectors are parallel or not. In order to check that the any two given vectors are parallel, then let say two vectors A and B are parallel if and only if they are scalar multiples of one another i.e. \[A~=~k\cdot B~\] , k is a constant not equal to zero. The alternative way to find out the two vectors are parallel is that the angle between those vectors must be equal to \[{{0}^{0}}\] .
Formula used:
Two vectors A and B are parallel if and only if they are scalar multiples of one another.
Condition under which vectors \[A~=\left( Ax,Ay \right)\] and \[B~=\left( Bx,By \right)\] are parallel is given by \[~\Rightarrow \dfrac{Ax}{Bx}~=~\dfrac{Ay}{By}~\]
Or
 \[\Rightarrow Ax\cdot By=Ay\cdot Bx\]

Complete step-by-step answer:
Two vectors A and B are parallel if and only if they are scalar multiples of one another.
 \[A~=~k\cdot B~\] , k is a constant not equal to zero.
Let \[A~=\left( Ax,Ay \right)\] and \[B~=\left( Bx,By \right)\]
A and B are parallel if and only if \[A~=~k\cdot B~\]
Therefore,
 \[\left( Ax,Ay \right)\text{ }=~k~\left( Bx,By \right)=\left( k\cdot Ax,k\cdot By \right)\]
Now,
 \[Ax=k\cdot Bx~\] And \[Ay=k\cdot By\]
Or
 \[~\dfrac{Ax}{Bx}~=~k~\] And \[\dfrac{Ay}{By}~=~k~\]
Condition under which vectors \[A~=\left( Ax,Ay \right)\] and \[B~=\left( Bx,By \right)\] are parallel is given by \[~\Rightarrow \dfrac{Ax}{Bx}~=~\dfrac{Ay}{By}~\]
Or
 \[\Rightarrow Ax\cdot By=Ay\cdot Bx\] .

Alternative method:
If two vectors are parallel, then angle between those vectors must be equal to \[{{0}^{0}}\]
We can find the cross product of both the vectors.
So,
 \[\sin 0={{0}^{0}}\]
Therefore, to show two vectors are parallel, then find the angle between them.

Note: Whenever such a type of question is given, where we need to find whether the given vectors are parallel or not. Find the cross products of the two vectors, if the cross product is equal to zero then the given vectors are parallel otherwise not. You can also use the condition that two vectors are parallel if and only if they are scalar multiples of one another otherwise they are not parallel.