Given two different non-parallel vectors, vec a and vec b, prove that vec a+vec b and vec a−vec b are the bisector vectors of the angle formed by vec a and vec b.

Sydney Stein

Sydney Stein

Answered question

2022-08-11

Given two different non-parallel vectors, a and b , prove that a + b and a b are the bisector vectors of the angle formed by a and b .

This question is in my book of vector geometry, but it is kind of weird for me. I do not know if I am approaching it the right way.
I know that the formula for the angle bisector vector is given like this:
Let u and v be vectors of non-zero length. Let u and v be their respective lengths. Then u v + v u is the angle bisector of u and v. So the only way a + b is the bisector vector is that both a and b have a length of one unit, but the question does not talk about this particular case, it seems phrased for all cases. Also I do not see how a - b could satisfy this formula.

Answer & Explanation

wietselau

wietselau

Beginner2022-08-12Added 28 answers

A bit of geometry:
In a Cartesian coordinate system
Consider a = A B , b = B C ..
Complete to form a parallelogram A B C D .
a + b = A C , a diagonal in the parallelogram .
1) C A B = D C B := α, since A B | | D C.
2) C A D = A C B := β, since A D | | B C.
A B C is isosceles , i.e. α = β

| | A B | | = | | B C | | , i.e. they have the same length.
traquealwm

traquealwm

Beginner2022-08-13Added 4 answers

You are rigth, if you don't have equality hipothesis ∥u∥=∥v∥ the conclusion is false in general, you can draw any two vectors with diferent length and see difente angles.

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