dot product

Question

|a|=3  |b|=5. find the value of β if a+βb and a-βb are perpendicular.

Answer 1

Since (a+βb) and (a-βb) are perpendicular, it means that (a+βb)•(a-βb)=0 or

 

a•a-β²*(b•b)=|a|²-β²*|b|²=0

 

Substituting values for |a| and |b| into the equation, we obtain:

 

9-25β²=0 or β=±3/5

Answer 2

if x and y are perpendicular vectors, then the angle θ between them is 90 degrees. The dot product x•y is defined as |x| |y| cos θ. Cos (90 degrees) = 0, therefore the dot product of perpendicular vectors is zero.

Therefore the dot product (a+βb)•(a-βb)= 0

 

9 - 25β² = 0

β² = 9/25

β = ±3/5