# Proving Angle Bisector of A and B with Vector Magnitudes

• thenewbosco
In summary, to prove that \frac{|B|A+|A|B}{|A|+|B|} is the bisector of the angle formed by A and B, you can construct a triangle with sides A, B, and B-A and use the cosine half angle formula to find the angle between A and B. Then, you can find the angle between A and the bisector vector C by taking the dot product of A and C and using the cosine formula. Similarly, you can find the angle between B and C. By comparing these angles, you can show that C is indeed the bisector of the angle formed by A and B.
thenewbosco
Prove that $$\frac{|B|A+|A|B}{|A|+|B|}$$ is the bisector of the angle formed by A and B. where i have used normal text for vector and abs value bars to represent magnitude of vector.

i have no clue how to get started on this. i have tried many approaches such as constructing a triangle with a, b, and b-a, but i cannot seem to make any progress. a couple of hints on getting started would be appreciated

Write two equations:

$$ax+by+c = 0$$
$$cx+dy+e = 0$$

and what do these equations represent?

I don't know off the top of my head (i'm not a vector geometry expert), but if you call the bisector vector C, taking $$A \cdot C$$ and $$B \cdot C$$ and knowing the cosine half angle formula should be a decent way to start

a couple of hints on getting started would be appreciated
You want to know that the vector you constructed (I'll call it C) is the angle bisector of A and B. Therefore, you want to know:

(1) The angle between A and B
(2) The angle between A and C
(3) The angle between B and C

don't you?

Suppose that A and B intersect at some point Q, and R is some point on A , and S is some point on B . Write the vector equations of the individual lines, and then of the bisector.

Last edited:
so i wrote A=Q+tQR
and B=Q+tQS,
as my two vector equations...how can i write the bisector

## 1. How do you prove that the angle bisector of A and B is true using vector magnitudes?

To prove that the angle bisector of A and B is true using vector magnitudes, you can use the formula for the angle between two vectors: cosθ = (a ⋅ b) / (|a| |b|). If the two vectors are equal, then the angle between them is bisected.

## 2. What are the steps for proving the angle bisector of A and B with vector magnitudes?

The steps for proving the angle bisector of A and B with vector magnitudes are as follows: 1. Identify the two vectors, A and B, that form the angle. 2. Calculate the magnitudes of the vectors, |A| and |B|. 3. Calculate the dot product of the two vectors, A ⋅ B. 4. Use the formula cosθ = (a ⋅ b) / (|a| |b|) to find the angle between the two vectors. 5. If the angle between the two vectors is half of the full angle, then the angle bisector is proven.

## 3. Can the angle bisector of A and B be proven with other methods besides vector magnitudes?

Yes, the angle bisector of A and B can also be proven using geometry and trigonometry. For example, you can use the angle bisector theorem, where the bisector of an angle divides the opposite side into two segments that are proportional to the adjacent sides of the angle.

## 4. Why is it important to prove the angle bisector of A and B with vector magnitudes?

Proving the angle bisector of A and B with vector magnitudes is important because it provides a mathematical and scientific basis for understanding the concept of an angle bisector. It also helps to establish a geometric relationship between the two vectors and the angle they form.

## 5. Are there any limitations to proving the angle bisector of A and B with vector magnitudes?

Yes, there are limitations to proving the angle bisector of A and B with vector magnitudes. This method only works for two-dimensional vectors and cannot be applied to three-dimensional or higher-dimensional vectors. Additionally, the vectors must be non-zero and non-parallel for the formula to work.

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