equation of a plane through a point and parallel to yz plane?

Question

I'm trying to find the equation of a plane going through a point (1,2,3) and parallel to the yz plane. To do this I know that I have to have three points to construct two vectors with them. Then do the cross product of the two vectors to find a normal vector to the plane, and then plug in the numbers of one point into the general equation of a plane. But....
I only have one point, so do I just pick any other two points that have any y and z coordinates but the same x coordinate to form a plane parallel to the yz plane?? for example (1, 4, 6) and (1, 8, -5) would that be right or would the numbers I pick change the equation of the plane that I'm trying to find? would the plane be "bounded" by the numbers I pick or would its equation be the same no matter what numbers I choose??? Is this the right way to go about solving this problem???
A thousand thanks if you can answer some or all of my questions! and 10 points to the best answer!
Thanks and please try to answer!, all related answers are welcome! :P

Answer 1

The equation of the y-z plane x = 0. So the vector

perpendicular to the y-z plane is simply <1, 0, 0>.

(Recall that the vector perpendicular to the plane

ax + by + cz = d is )

Now use the general formula:



1(x - 1) + 0(y - 2) + 0(z - 3) = 0



this gives (x - 1) = 0, which is x = 1



So the equation of the plane is x = 1.

Answer 2

Equation of xz-plane: y = 0 -----> 0x + 1y + 0z = 0 -----> normal = < 0, 1, 0 > Equation of yz-plane: x = 0 -----> 1x + 0y + 0z = 0 -----> normal = < 1, 0, 0 > Vector that is orthogonal to both < 0, 1, 0 > and < 1, 0, 0 > will be parallel to both planes To find this vector, take cross product: < 1, 0, 0 > x < 0, 1, 0 > = < 0, 0, 1 > Since vector < 0, 0, 1 > is parallel to both planes, then we use this as direction vector of line: r(t) = (2, 3, 4) + t (0, 0, 1)

Answer 3

You have actually overcomplicated it. The yz plane is simply x=0 so the plane that is parallel to the yz plane that passes through point (1,2,3) is simply x=1. Now you could work it out numerically but this problem is really more conceptual, and therefore much easier to work conceptually.