# Angle between two planes with two common points and orientation

Given are four points in three-dimensional space: P1, P2, P3 and P4. From this I form the plane through P1,P2 and P3 and the plane plane P2, P3 and P4. Using the formula cos θ = / ( || N1|| . ||N2 || ) I get the angle between these two planes.

However, I would like to know not only the value of the angle, but also the sentence: when I look from point P3 to P2, is P1 to the left or right of P4? How can I calculate this?

Thank you in advance!

Asker: Joey, 26 years old

## Answer

Dear Jody,

When determining an oriented angle, we must realize that the point of view from which we view the situation is very important. Just to look at your last question: If I look from point P3 to P2, is P1 to the left of P4? You would get a reversed answer if you turn your head and rotate 180 degrees around the axis P2-P3. So we have to be very careful and formulate the question well.
I hope I understand your question correctly if I translate them to: if I look from P2 to P3 and I call α1 the half-plane in which P1 lies and α2 the half-plane that contains P4, what is the angle (between -180 degrees and 180 degrees) that is made between α1 and α2? Here, a positive angle corresponds to counterclockwise rotation.
A convenient method for determining an oriented angle is to use the vectorial product of 2 vectors (also known as the cross product).
The vector product v ∧ w of 2 vectors v and w is the vector perpendicular to v and perpendicular to w, such that v, w and v ∧ w satisfy the right-hand rule. 3 (lin. independent) vectors a,b,c satisfy the right-hand rule if the orientation of the vectors is correct when vector a is compared to the thumb, vector b is compared to the index finger, and vector c is compared to the middle finger ( all of the right hand). (This rule is also called the corkscrew rule).
In the case of your problem, we then have to compare the vector product of the vectors N1 and N2 with the vector P2P3. It is important that you know the orientation of your N1 and N2.
A possible scenario is the following (with names as in the figure).
Determine N1 as (P2P1 ∧ P2P3) / ||P2P1 ∧ P2P3|| and N2 as (P2P4 ∧ P2P3) / || P2P4 ∧ P2P3 ||.
The angle θ then corresponds to the oriented angle from N1 to N2. Furthermore, N1 ∧ N2 = k*P1P2. If k is positive it will be a positive angle (clockwise), if k is negative it will be a negative angle.
In the example, N1 ∧ N2 will point in the opposite direction as P2P3 (after all, use the right-hand rule) and the angle θ is given counterclockwise, if you look from P2 to P3.
With best regards,
Steve Symens

## Answered by

#### dr. Stijn Symens

Math

University of Antwerp
Prinsstraat 13 2000 Antwerp
http://www.uantwerpen.be

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