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Find a bisecting angle

  • Thread starter Calpalned
  • Start date
  • #1
297
6

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?
 

Answers and Replies

  • #2
33,287
4,990

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.
Calpalned said:
I would also like to confirm:

- if (a dot a) is always equal to |a|^2
Yes.
Calpalned said:
- 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.
Calpalned said:
- 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.
 
  • #3
33,287
4,990
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.
 

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