Discrete Math: Finding Angle Between Plane & XZ Axis

[eme_girl]

How do you find the angle between the co-ordinate axis (i.e. the xz plane) and another plane in general?

 

[learningphysics]

How do you find the angle between the co-ordinate axis (i.e. the xz plane) and another plane in general?

Angle between two planes is the angle between the normals of the planes.

 

[HallsofIvy]

Expanding on what learningphysics said, if one plane is given by Ax+ By+ Cz= D and the other by ax+ by+cz= d, then the normal vectors are Ai+ Bj+ Ck and ai+ bj+ ck respectively. u.v= |u||v|cos(θ) so θ, the angle between the two vectors and the angle between the planes, is given by cos(θ)= u.v/(|u||v|).


In particular, the xz-plane has normal vector j. If the other plane is given by Ax+By+Cz= D, its normal vector is Ai+Bj+Ck. The dot product of those is simply B so the angle between the planes is given by $cos(\theta)= \frac{B}{\sqrt{A^2+B^2+C^2}}$.