piopiopioirp

2022-11-23

Can I find length of bisector by knowing the position vectors?
$\stackrel{\to }{A}$
$\stackrel{\to }{B}$
$\stackrel{\to }{C}$
To find the length of angle bisector of bac I marked the points A(1,−1,−3) B(2,1,−2) C(−5,2,−6). How can I use the fact that the angle between the bisector and two adjacent sides is equal?

mentest91k99

Expert

Method 1Just find the lenth of the sides of the triangle amd use the formula $\frac{2\sqrt{bcs\left(s-a\right)}}{b+c}$
Method 2: By the angle bisector theorem we know the ratio in which the bisector divides the side opposite to the bisected angle. Use the section formula too find point of intersection and then use distance formula

ajakanvao

Expert

$\stackrel{\to }{AB}=\stackrel{\to }{b}-\stackrel{\to }{a}={\stackrel{\to }{r}}_{1},\stackrel{\to }{AC}=\stackrel{\to }{c}-\stackrel{\to }{a}={\stackrel{\to }{r}}_{2}$ then position vector of the angle bisector od BAC angle is
$\stackrel{\to }{r}=\frac{{\stackrel{^}{r}}_{1}+{\stackrel{^}{r}}_{2}}{2}$
The length of angle bisector is $|\stackrel{\to }{r}-\stackrel{\to }{a}|.$

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