If ` vec a ,\ vec b` are two vectors such that `| vec a+ vec b|\ =| vec b|,` then prove that ` vec a+2 vec b` is perpendicular to ` vec adot`

Explanation

Given that,
`| vec a+ vec b|=| vec b|`
squaring on both sides we get,
`=>|vec a+ vec b|^2 = |vec b|^2`
`=>(vec a+ vec b)*(vec a+ vec b)=|vec b|^2`
`=>|vec a|^2+|vec b|^2+2 vec a vec b= |vec b|^2`
`=>|vec a|^2+2 vec a vec b=0`
`(vec a+ 2 vec b)* vec a`
`=|vec a|^2+2 vec a vec b`
`=0`
hence `(vec a+ 2 vec b) bot vec a`