If |a+b|=|a-b|,show that a and b are perpendicular vectors

Asked by Snehil Sankalp | 20th Apr, 2012, 12:16: PM

Expert Answer:

Given: |a+b|=|a-b|
To Prove: a and b are perpendicular vectors
 
So, if you square both the sides, then you will get 
 
(|a+b|)^2 = (|a-b|)^2
 
Since, any |a|^2 can be written as the dot product of a with itself i.e. a.a , so here also (|a+b|)^2 can be represented as (a+b).(a+b). Using the same logic 
 
(a+b).(a+b) = (a-b).(a-b)
On expanding, we will get
a.a +b.b +2a.b = a.a +b.b -2a.b
4a.b=0
4|a|*|b|*cos(theta) = 0 
Since, a and b vectors are non-zero vectors, hence
cos(theta) = 0
and hence, theta = 90
 
So, the angle between a and b vectors is 90. 
 
Thank you

Answered by  | 21st Apr, 2012, 10:47: AM

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