Planes and parametric equations

[grassstrip1]

Hi everyone! I'm having some issues with this problem for linear algebra. I understand parametric equations fairly but I'm confused about the unit vector notation

1) Consider the plane r(s,t)=2i + (t-s) j + (1+3s-5t) k find the z component of the point (2,-1, z₀)

For what values of s and t is this the case?

I don't really know how to start the problem because it isn't in vector or parametric form like I'm used to.

 

[Mark44]

Hi everyone! I'm having some issues with this problem for linear algebra. I understand parametric equations fairly but I'm confused about the unit vector notation

1) Consider the plane r(s,t)=2i + (t-s) j + (1+3s-5t) k find the z component of the point (2,-1, z₀)

For what values of s and t is this the case?

I don't really know how to start the problem because it isn't in vector or parametric form like I'm used to.

It's easy enough to get from the vector form to the parametric form of this plane.
Here x = 2, y = t - s, and z = 1 + 3s - 5t, and you're given a point (2, -1, z₀).

 

[HallsofIvy]

I don't know what you are "used to" but it certainly is in "vector form" and, as Mark44 says, it is easy to convert to parametric form:
x= 2, y= t- s, z= 1+ 3s- 5t. In order to have (x, y, z)= (2, -1, z_0) you must have 2= 2, t- s= -1, and 1+ 3s- 5t= z_0.

Perhaps it is the fact that there is not a single "unique" answer that is bothering you?

There are an infinite number of points, in fact an entire line, with x= 2, y= -1. From t- s= -1, we can get t= s+ 1 and so write z_0= 1+ 3s- 5(s+ 1)= 1+ 3s- 5s- 5= -4- 2s. The set of such points consists of the line x= 2, y= -1, z= -4- 2s, for any s.

 

[grassstrip1]

Thank you for the replies! I left a little something out of the problem, it said find the z component so that it lies on the plane. Wouldn't that make it just one specific point?

 

[Mark44]

Thank you for the replies! I left a little something out of the problem, it said find the z component so that it lies on the plane. Wouldn't that make it just one specific point?

Work the problem through and see.