Question

How to show whether two vectors are parallel?

Answer 1

Hint: Two vectors A and B (say) are parallel if and only if they are scalar multiples of one another, i.e., $A=kB,k$ is a constant not equal to zero or if the angle between the vectors are equal to $0^{\circ }$.

Complete step-by-step answer:
Let us assume two vectors $\overrightarrow{u}$ and $\overrightarrow{v}$.

To prove the vectors are parallel-

Find their cross product which is given by, $\overrightarrow{u}\times \overrightarrow{v}=|u||v|\sin \theta$.

If the cross product comes out to be zero.

Then the given vectors are parallel, since the angle between the two parallel vectors is $0^{\circ }$ and $\sin 0^{\circ }=0$.

If the cross product is not equal to zero then the vectors are not parallel.

Another way could be is to find if one vector is scalar multiple of the second vector or not,

If $u=ku,k$ is a constant and $k\ne 0$, then the vectors u and v will be parallel.

If $u\ne ku$, then vectors u and v are not parallel.

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 2 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.