How do you find the vector parametrization of the line of intersection of two planes #2x - y - z = 5# and #x - y + 3z = 2#?

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Answer 1 · 2015-03-22T12:18:52

One answer could be:

#x=t#

#z=1/4t-3/4#

#y=7/4t-17/4#.

Let's solve the system of the two equations, explaining two letters in function of the third:

#2x-y-z=5#

#x-y+3z=2#

So:

#y=2x-z-5#

#x-(2x-z-5)+3z=2rArrx-2x+z+5+3z=2rArr#

#4z=x-3rArrz=1/4x-3/4#

so:

#y=2x-(1/4x-3/4)-5rArry=2x-1/4x+3/4-5#

#y=7/4x-17/4#.

SO:

#z=1/4x-3/4#

#y=7/4x-17/4#

and if we put #x=t#:

#x=t#

#z=1/4t-3/4#

#y=7/4t-17/4#.