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Let a and b be two vectors such that neither of a-b and a+b is the zero vector.

(a) Prove that if the vectors a and b are of equal length, then a-b and a+b are perpendicular.

(b) Prove that if a-b and a+b are perpendicular, then a and b are of equal length.

 May 7, 2024
 #1
avatar+129712 
+1

(a)

 

Let  a  = ( 0, m)

Let b = (m , 0)

Note that these are equal in length

 

(a + b)  =  (m , m)

(a - b)  =  (-m , m)

 

If  two vectors are perpendicular, their dot product  =  0

 

dot product =   (m , m)  ( dot )  (-m , m)  = -m^2  + m^2  =   0

 

 

cool cool cool

 May 7, 2024
 #2
avatar+68 
+1

How did you get a  = ( 0, m) and b = (m , 0). Why the 0?

LearningCat  May 7, 2024
 #3
avatar+129712 
0

Think about this...let m  =  1

 

So   a  =  (1,0)    and b =(0, 1)

 

These are the same  length  and  are perpendicular

 

cool cool cool

CPhill  May 7, 2024

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