How do I find the angle between the planes #x + 2y − z + 1 = 0# and #x − y + 3z + 4 = 0#?

1 Answer
Rui D.
Jan 4, 2016

#alpha=119.50^@#

Explanation:

To find the angle between two planes, one has to find the angle between normal vectors of these planes.

How to find a normal vector to a plane

Explanation of why the angle between two vectors, each one normal to a plane, gives the angle between the two involved planes

Normal Vectors
For the plane 1 , #x+2y-z+1=0#
# N vec 1= hat i +2*hat j-1*hat k#

For the plane 2 , #x-y+3z+4=0#
# N vec 2= hat i -hat j+3*hat k#

Angle between the 2 vectors
#N vec 1*N vec 2=|N vec 1|*|N vec 2|*cos alpha# => #cos alpha=(N vec 1*N vec 2)/(|N vec 1|*|N vec 2|)#
#N vec 1*N vec 2=1*1+2*(-1)+(-1)(3)=1-2-3=-4#
#|N vec 1|=sqrt(1^2+2^2+(-1)^2)=sqrt(1+4+1)=sqrt(6)#
#|N vec 2|=sqrt(1^2+(-1)^2+3^2)=sqrt(1+1+9)=sqrt(11)#
#cos alpha=-4/(sqrt(6)*sqrt(11)# => #cos alpha=-4/sqrt(66)# => #alpha=119.50^@#

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