Prove c bisects angle between a and b | Vector Algebra

[Calpalned]

Homework Statement

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.

Homework Equations

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?

 

[Mark44]

Homework Statement

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.

Homework Equations

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?

This doesn't look right. If θ is the angle between a and c, you should have $cos(θ) = \frac{a \dot c}{|a||c|}$. When you substitute in c and |c|, there should be some stuff in the denominator. I don't see that in what you have.

I would also like to confirm:

- if (a dot a) is always equal to |a|^2

Yes.

- how to differentiate absolute value and magnitude as they use the same symbol

Absolute values apply to real numbers. For a vector, |v| means the magnitude of the vector.

- When I got (2|a||c|)(a dot b), do I do regular multiplication or use the dot product?

Regular multiplication, if I'm understanding what you are asking. The two quantities in parentheses are real numbers.

 

[Mark44]

A simpler approach than you're taking is to calculate the cosines of the two angles; i.e., the angle between a and c, and the angle between c and b. As expected, these turn out to be equal.