# Why do we get angle between planes by taking dot product of the normal vectors?

Let theta be the angle between 2 planes. Then to find this angle we take the dot product of the two normal vectors to the plane, divide by their magnitudes and then finally take the arccos of the resulting value. However, after doing this why do we not get 180 - theta and rather get theta? angle between two planes

• You can get $180-\theta$ if you take one of the vectors in the opposite direction. – John Doe May 3 '17 at 18:11
• okay but why do we get theta in the first place? isn't the angle between the two normal vectors 180 - theta? – abc May 3 '17 at 18:14
• It depends on what you mean by "the angle between the planes." There are two angles, after all, one $\leq 90$ degrees and another $\geq 90$ degrees. We usually take the smaller of the two, which is also the smaller of the two angles between the normal vectors. – kccu May 3 '17 at 18:17
• @abc note that there are two supplementary angles that can be called "the angle between planes". Similar to the idea of an angle between two lines. – Omnomnomnom May 3 '17 at 18:17
• I understand there are two supplementary angles. But if you refer to the link showing the diagram, why do we get theta rather than 180 - theta? Isn't 180 - theta the angle between the vectors considering the direction in which they are pointing. @Omnomnomnom – abc May 3 '17 at 18:22