Question

If `veca and vecb` are two vectors such that `|veca+vecb|=|veca|,` then prove that vector `2veca+vecb` is perpendicular to vector `vecb`

 

Answer 1

Given |`veca+vecb|=|veca|`∣

`|veca+vecb|^2=|veca|^2`

`|veca|^2+2veca.vecb+|vecb|^2=|veca|^2`

`2veca.vecb+|vecb|^2=0 ................(1)`

`Now (2veca+vecb)(vecb)=2vecavecb+vecbvecb=2vecavecb+|vecb|^2=0  using(1)`

We know that, if the dot product of two vectors is zero, then either of the two vectors is zero or the vectors are perpendicular to each other.

Thus,

Given `|veca+vecb|=|veca|`

`|veca+vecb|^2=|veca|^2`

`|veca|^2+2veca.vecb+|vecb|^2=|veca|^2`

`2veca.vecb+|vecb|^2=0 ................(1)`

`Now (2veca+vecb)(vecb)=2vecavecb+vecbvecb=2vecavecb+|vecb|^2=0  using(1)`

We know that, if the dot product of two vectors is zero, then either of the two vectors is zero or the vectors are perpendicular to each other.

Thus, `2veca+vecb` is perpendicular to `vecb`