# Thread: Finding the angle between to vectors

1. ## Finding the angle between to vectors

im trying to Find the angle between to vectors, when they are placed tail to tail
how do you do this...(in 2D and 3D)...im trying to use the law of cosines, manipped so cos(theta) is on one side......but im not getting the right answer

[cos19, sin19] and [cos54, sin54]

2. Originally Posted by stones44
im trying to Find the angle between to vectors, when they are placed tail to tail
how do you do this...(in 2D and 3D)...im trying to use the law of cosines, manipped so cos(theta) is on one side......but im not getting the right answer

[cos19, sin19] and [cos54, sin54]
use the dot product:

$\displaystyle u \cdot v = |u||v| \cos \theta$

where $\displaystyle u$ and $\displaystyle v$ are vectors and $\displaystyle \theta$ is the angle between them, this works in 2D and 3D

3. ok thanks

now to find |u| or |v| in 3D

so..

[a,b,c]

root(a^2 + b^2) =x

root(x^2 + c^2) =|u|

right?

4. $\displaystyle u = \left\langle {a,b,c} \right\rangle \quad \Rightarrow \quad \left\| u \right\| = \sqrt {a^2 + b^2 + c^2 }$

If u & v are vectors, the angle between them is
$\displaystyle \phi = \arccos \left( {\frac{{u \cdot v}}{{\left\| u \right\|\left\| v \right\|}}} \right)$