Find the rectangular equation for the surface by eliminating parameters from the vector-valued function. Identify the surface. $\overrightarrow { \mathbf { r } } ( u , v ) = u \hat { \mathbf { i } } + v \hat { \mathbf { j } } + \frac { v } { 10 } \hat { \mathbf { k } }$
A) the plane $z = \frac { x } { 10 }$
B) the cylinder $x ^ { 2 } + y ^ { 2 } = 100$
C) the plane $x = \frac { y } { 10 }$
D) the plane $z = \frac { y } { 10 }$
E) the plane $y = \frac { x } { 10 }$

Question

Quizplus · Accepted Answer

To eliminate parameters, we need to solve for x, y, and z in terms of u and v.
From the given vector-valued function, we have:
x = u
y = v
z = v/10

Solving for v in terms of z, we get:
v = 10z

Substituting this into the expressions for x and y, we get:
x = u
y = 10z

Therefore, the rectangular equation for the surface is z = y/10, which corresponds to choice D.
This is a plane that intersects the y-axis at 0 and has a slope of 1/10 in the xz-plane.

Find the Rectangular Equation for the Surface by Eliminating Parameters r→(u,v)=ui^+vj^+v10k^\overrightarrow { \mathbf { r } } ( u , v ) = u \hat { \mathbf { i } } + v \hat { \mathbf { j } } + \frac { v } { 10 } \hat { \mathbf { k } }r(u,v)=ui^+vj^​+10v​k^

Find the Rectangular Equation for the Surface by Eliminating Parameters $\overrightarrow { \mathbf { r } } ( u , v ) = u \hat { \mathbf { i } } + v \hat { \mathbf { j } } + \frac { v } { 10 } \hat { \mathbf { k } }$