Question

If |a-b| = 1/2|b| and (a-b) is perpendicular to a then what is the angle between a and b

Answer 1

Use definition of dot-product.

(A-B).B = 0 because (A-B) and B are perpendicular.

A.B-B.B=0

|A|*|B|*cos(alpha) - |B|^2 = 0

Now we use the fact that |A|=2|B|

2*|B|*|B|*cos(alpha) - |B|^2 = 0

2*|B|^2*cos(alpha) -|B|^2=0

cos(alpha) = 1/2

alpha = pi/3.

Answer 2

Answer: as they have said that (a-b) is perpendicular to a do a dot product b/w (a-b) and a. after that substitute the result into the first statement. on ongoing solving you will arrive to the solution.

Explanation:

[a-b]=1/2[b]

a²+b²-2abcos(α)=1/4b²_____(1)

(a-b).a=0

a .a - a .b=0

a²-abcos(α)=0

a²=ab cos(α)_____________(2)

substitute 2 into 1

on solving we get α=π/6=30°