Parametric Surfaces Homework Help

[goonking]

Homework Statement

Homework Equations

The Attempt at a Solution

so to start this off, I choose a random point, by setting u and v = 0

giving me the point (0,3,1) but I have no idea how what to do next.

how do I find ua and vb?

 

[tommyxu3]

Thinking directly, $a$ and $b$ must be to vectors lying on the plane. Maybe you can set they start at $r_0,$ that is your $(0,3,1)$ and find any two independent vector to structure the plane.

 

[goonking]

Thinking directly, $a$ and $b$ must be to vectors lying on the plane. Maybe you can set they start at $r_0,$ that is your $(0,3,1)$ and find any two independent vector to structure the plane.

sorry, a bit confused. do I plug in more numbers for v and u?

 

[tommyxu3]

$v$ and $u$ then are parameter. Selected $a$ and $b$ will dominate the form of the plane.

 

[Ray Vickson]

Homework Statement

View attachment 89508

Homework Equations

View attachment 89509

The Attempt at a Solution

so to start this off, I choose a random point, by setting u and v = 0

giving me the point (0,3,1) but I have no idea how what to do next.

how do I find ua and vb?

If $(x,y,z)$ are the cartesian coordinates of a point on the surface, how do you express the values of $x$, $y$ and $z$ in terms of the parameters $u$ and $v$? Can you use those expressions to re-write the surface in the form $z = a + b x + cy$?

 

[goonking]

If $(x,y,z)$ are the cartesian coordinates of a point on the surface, how do you express the values of $x$, $y$ and $z$ in terms of the parameters $u$ and $v$? Can you use those expressions to re-write the surface in the form $z = a + b x + cy$?

how did you think of the form $z = a + b x + cy$? does the surface have to be in that form?

 

[goonking]

how did you think of the form $z = a + b x + cy$? does the surface have to be in that form?

anyway, the textbook came up with

= <0,3,1> + u<1,0,4> + v<1,-1,5> and I have no idea how they came up with u<1,0,4> and v<1,-1,5>.

how are they coming up with vectors with just a given point?!

 

[Ray Vickson]

anyway, the textbook came up with

= <0,3,1> + u<1,0,4> + v<1,-1,5> and I have no idea how they came up with u<1,0,4> and v<1,-1,5>.

how are they coming up with vectors with just a given point?!

What are $\bf{i}, \bf{j}$ and $\bf{k}$?

 

[goonking]

What are $\bf{i}, \bf{j}$ and $\bf{k}$?

i = <0,3,1>
j= u<1,0,4>
k=v<1,-1,5>

 

[tommyxu3]

They are unit vectors on the three dimension instead of what you say. The solution makes it to the form to match the required.