Proof that Vector v Bisects the Angle Created by Vectors a and b

In summary, to show that vector v bisects the angle created by vectors a and b, we can use the dot product equation to find the angle between a and v. By substituting ((|a|b + |b|a) / (|a| + |b|)) for v, we can show that the angles formed by a and v and b and v are equal, thus proving that v bisects the angle. This can also be shown algebraically by adding two different vectors of equal lengths, which results in a third vector whose direction bisects the angle of the first two vectors.
  • #1
meson0731
14
0

Homework Statement



Given vector a and b, let vector v = (|a|b + |b|a) / (|a| + |b|). Show that vector v bisects the angle created by vector b and a.

Homework Equations



cos(θ) = a dot b / (|a||b|)


The Attempt at a Solution



I used the dot product equation to find angle between a and v.

cosθ = (a dot v) / |a||v|

i substituted ((|a|b + |b|a) / (|a| + |b|)) for v.

cosθ = (a dot ((|a|b + |b|a) / (|a| + |b|)) / |a||v|

I distributed the (a dot ((|a|b + |b|a)

cosθ ( ((|a|b dot a) + (|b| |a|^2) ) / (|a| + |b|)) / (|a||b|)

I then tried to get the cosθ of b dot w to the same thing. I tried to get the two angles equal to each other. My idea is that if the angles formed by a and v and b and v are equal then v bisects the two angles.
 
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  • #2
So you have
$$\cos\theta_{av} = \frac{a(\vec{a}\cdot\vec{b})+a^2b}{(a+b)av}$$ where ##a=|\vec{a}|##, ##b=|\vec{b}|##, and ##v=|\vec{v}|##, and the corresponding expression for ##\cos\theta_{bv}##. So far so good. Try calculating
$$\frac{\cos\theta_{av}}{\cos\theta_{bv}}$$ and show it's equal to 1.
 
  • #3
Meson,

The problem becomes easy when you realize that if you add two different vectors of equal lengths, then you get a third vector whose direction bisects the angle of the first two vectors. Try it and see. That takes care of the geometry. Now see if you can figure out the algebra.

Ratch
 
Last edited:

1. What is meant by a vector bisecting an angle?

A vector bisecting an angle means that it divides the angle into two equal parts.

2. How can I prove that a vector v bisects the angle created by vectors a and b?

To prove that a vector v bisects the angle created by vectors a and b, you can use the properties of vector addition and subtraction. If the sum of vectors a and v is equal to the sum of vectors b and v, then v bisects the angle between vectors a and b.

3. What is the importance of proving that a vector bisects an angle?

Proving that a vector bisects an angle is important in understanding and solving problems related to vector geometry. It is also a fundamental concept in mathematics and physics.

4. Can a vector bisect multiple angles at the same time?

Yes, a vector can bisect multiple angles at the same time as long as it intersects the angles at their midpoints.

5. What are some real-life applications of vectors bisecting angles?

Vectors bisecting angles are commonly used in navigation and surveying, as well as in engineering and architecture for creating accurate geometric designs and structures. They can also be applied in computer graphics for creating 3D models and animations.

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