Calculus III: Lines, planes, and surfaces in space. Find the angle and line of intersection between two planes.

Question

stamp · Answer

Find the angle and line of intersection between$$5x-3y+z=0$$and$$x+4y+7z=1$$
@khoala4pham @B25

stamp · Answer

Find the normal vector of each plane and use the dot product to find the angle between?

anonymous · Answer

That is a rather interesting idea. I can't be sure but it sounds plausible.

anonymous · Answer

Law of cosines

stamp · Answer

$$Vn_1=<5,-3,\ 1>$$$$Vn_2=<1,\ 4,\ 7>$$$$\theta =cos^{-1}(\frac{Vn_1•Vn_2}{|Vn_1||Vn_2|})$$$$Vn_1Vn_2=5-12+7=0$$$$\theta=\pi/2$$The planes are perpendicular?

anonymous · Answer

That was my dilemma. If you extract two lines L1 and L2 from the respective planes and find their normals N1 and N2, in 2 dimensions, these vectors form a quadrilateral. The fact that L1 N1 are orthogonal and L2 and N2 are orthogonal means that 180 degrees has already been used up. Do you see the problem? I find it hard to explain...

stamp · Answer

|dw:1360388376489:dw|

anonymous · Answer

Yes, that is one specific instance in which it works. However, if you can visualize it, if the two planes are sufficiently close enough--if they form a small enough angle--, the angle of the vectors n1 and n2 will not be the angle of the intersecting planes.

stamp · Answer

@khoala4pham We are going to go to bed and digest the material covered today. Thank you again for your time and instruction.

anonymous · Answer

@stamp I apologize. You are right. http://www.jtaylor1142001.net/calcjat/Solutions/VPlanes/V2PTheta.htm There must be something wrong with my thought process.

anonymous · Answer

I see it now. You are absolutely right, again I apologize. If you are interested, this website has an excellent diagram. http://www.netcomuk.co.uk/~jenolive/vect14.html

agent0smith · Answer

^ nice. That's the method i had in mind, but wasn't 100% sure it was valid. The diagrams are helpful.