Cartesian equation of the plane

Homework Statement

If the line "l" is given by the equations 2x-y+z=0, x+z-1=0, and if M is the point (1,3,-2), find a Cartesian equation of the plane.
a) passing through M and l
b) passing through M and orthogonal to l

(r-r0)n=0

The Attempt at a Solution

I expanded the equation above, so i got rn=r0n
then, (x,y,z)n=(1,3,-2)n.
And i don't know how to get the normal vector from those two equations

Last edited:

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jedishrfu
Mentor
can you be more specific? does the plane contain the line or the point? or does it contain both?

sorry, i forgot some of them.

a) passing through M and l
b) passing through M and orthogonal to l

jedishrfu
Mentor
and what course? vector algebra?

also you really need to show more work before anyone here at PF will help.

HallsofIvy
Homework Helper

Homework Statement

If the line "l" is given by the equations 2x-y+z=0, x+z-1=0, and if M is the point (1,3,-2), find a Cartesian equation of the plane.
a) passing through M and l
I presume you know: If <A, B, C> is a vector normal to a plane and $(x_0, y_0, z_0)$ is a point in the plane, then the Cartesian equation of the plane is $A(x- x_0)+ B(y- y_0)+ C(z- z_0)= 0$.

2x- y+ z= 0, x+ z=1. Subtract the second equation from the first to eliminate z: x- y= -1 or x= y- 1. If you let y= 0, x= -1 so x+ z- 1= -1+ z- 1= 0 gives z= 2. (1, 0, 2) is a point on the given line. If you let y= 1 in x= y- 1, x= 0 so x+ z- 1= z- 1= 0 gives z= 1. (0, 1, 1) is a point on the given line. Those, together with (1, 3, -2) give you three points in the plane. <1- 1, 3- 0, -2- 2>= <0, 3, -4> is a vector in the plane. <1- 0, 3- 1, -2- 1>= <1, 2, -3> is another vector in the plane. Their cross product is normal to the plane.

b) passing through M and orthogonal to l
<2, 1, 1> is a vector normal to 2x- y+ z= 0. <1, 0, 1> is a vector normal to x+ z= 1. Their cross product is normal to both so normal to any plane perpendicular to the line. You have a normal vector and a point in the plane.

(r-r0)n=0

The Attempt at a Solution

I expanded the equation above, so i got rn=r0n
then, (x,y,z)n=(1,3,-2)n.
And i don't know how to get the normal vector from those two equations

Last edited by a moderator: