If c = |a|b + |b|a, where a, b, and c are all nonzero vectors, show that c bisects the angle between a and b.
Angle between a & b is cos-1(a dot b)/(|a||b|)
Angle between a & c is cos-1(a dot c)/(|a||c|)
Angle AC is half of angle AB
The Attempt at a Solution
Given c = |a|b + |b|a, I plug that into the equation for the angle between a and c. I eventually get (|a||b|)(|a|b + |b|a2 = (2|a||c|)(a dot b). Is this right? I would also like to confirm:
- if (a dot a) is always equal to |a|^2
- how to differentiate absolute value and magnitude as they use the same symbol
- When I got (2|a||c|)(a dot b), do I do regular multiplication or use the dot product?