Question

Determine the line of intersection between
pi1: 4x+2y+6z-14=0
pi2: x-y+z-5=0

Answer 1

Not sure if there is a solid method, but what im thinking is if we can find 2 points on the line of intersection, then we can come up with a parametric equation of the line.

Answer 2

So basically we should pick a value for one of the variables, say z, and solve for the other 2. If we pick z = 0, we have the system:\[4x+2y=14\]\[x-y=5\]which yields the solution x = 4, y = -1. So (4,-1,0) is on this line. Now we need to do this again picking a different value of z.

Answer 3

Let z = 3 this time. This will produce the system:\[4x+2y=-4\]\[x-y=2\]which has the solutions x = 0 y = -2. So the other point is (0,-2,3). Now that we have those two points on the line, it should be easy to come up with the parametric equation of the line of intersection.

Answer 4

oh ok, thanks
can i just equal both lines together.. and eliminate one of the variable?

Answer 5

Let\[P_1=(4,-1,0),P_2=(0,-2,3)\]Then the vector from P1 to P2 is:\[(0-4)\hat{i}+(-2+1)\hat{j}+(3-0)\hat{k}=\langle-4,-1,3\rangle \]So the equation of the line that is the intersection of the two planes is:\[x=4-4t,y=-1-t,z=3t\]

Answer 6

yes. You want to think of it as a system of equations.

Answer 7

There is probably a faster way to do this, i just dont see it atm >.<

Answer 8

lol ok thanks for ur help !