Question

2
32. If A and B are two non-zero vectors having equal magnitude,
the angle between the vectors A and A - B is​

Answer 1

Answer:

Since ||A|| = ||B||, we know that the triangle formed by A, B and A-B must be isosceles.

Now let’s define A=(a1, a2, a3) and B=(b1, b2, b3). A.B = a1 b1 + a2 b2 + a3 b3. The dot product is also the cosine of the angle between A and B so we know the angle between A and B is given by acos(a1 b1 + a2 b2 + a3 b3).

This means the rest of the angles in the triangle bounded by A, B and A-B must be given by pi - acos(a1 b1 + a2 b2 + a3 b3) (in radian measure) or

180° - acos(a1 b1 + a2 b2 + a3 b3) if you’re using degrees.

Since the triangle is isosceles, it’s base angles are equal and therefore the angle at the base is 0.5 [pi - acos(a1 b1 + a2 b2 + a3 b3)] (in radian measure) or

0.5[180° - acos(a1 b1 + a2 b2 + a3 b3)] if you’re using degrees