How do I find the angle between these two vectors

a = [-1, 1, 1]
b = [3, -2, 2]

Dot Product is -3
Modulus of A is 3
Modulus of B is 17.

3 * 17 = 51
sqrt of 51 = 7.141
1 / 7.141 = 0.140 * - 3 = -2.860

I dont know where to go from here. Usually I get a number like 0.xxxxxxxx and I find the cos θ of that number but its not working with a negative number like this one.
I dont know how to find the angel of a negative number

The cosine of the angle is equal to the dot product / (||A|| ||B||). So:

cos θ = -3 / sqrt(3*17) ~=-0.42

θ = 114 degrees.


Your method was exactly correct. It looks like you made a slight calculator error in your final step:

"0.140 * - 3 = -2.860" is wrong; it's -0.42
0.140 - 3 = -2.860 though ...

Ed: Excuse me - it was 114.8 degrees, closer to 115.

For addtional information, please see also tha following data links:

http://en.wikipedia.org/wiki/Vectors_(ma…
http://chemistry.about.com/od/workedchem…
http://www.wikihow.com/Find-the-Angle-Be…
http://math.stackexchange.com/questions/…
http://www.euclideanspace.com/maths/alge…

a • b = IaI IbI cos Ө

- 3 - 2 + 2 = √3 √17 cos Ө

cos Ө = - 3 / √51

Ө = 180 - 65.1 °____ is acceptable

Ө = 114.9°