Find a vector function that represents the curve of intersection of the two surfaces. The cylinder x^2+y^2=4 and the surface z=xy

Efan Halliday

Efan Halliday

Answered question

2021-05-25

Find a vector function that represents the curve of intersection of the two surfaces. The cylinder x2+y2=4 and the surface z=xy

Answer & Explanation

estenutC

estenutC

Skilled2021-05-26Added 81 answers

Solving the equation with a vector:

image

Jeffrey Jordon

Jeffrey Jordon

Expert2021-09-30Added 2605 answers

Parametric equations for x and y for a circle

x=r×cos(t)

y=r×sin(t)

Because the formula is x2+y2=4 the radius is 4=2. Then plug in 2 for the parametric equations

x=2cos(t)

y=2sin(t)

Because the surface is z=xy, substitute x and y for their respective parametric equations, ergo creating a parametric equation for z.

z=[2cos(t)][2sin(t)]

Simplify

z=4sin(t)cos(t)

Final answer:

<2cos(t),2sin(t),4sin(t)cos(t)>

alenahelenash

alenahelenash

Expert2023-05-10Added 556 answers

To find a vector function that represents the curve of intersection between the cylinder x2+y2=4 and the surface z=xy, we can parameterize the curve using a parameter t.
Let's start by considering the equation of the cylinder:
x2+y2=4
We can express x and y in terms of a trigonometric function, such as sine and cosine, to simplify the equation. Let's choose x=2cos(t) and y=2sin(t), where t ranges from 0 to 2π.
Substituting these values into the equation of the surface z=xy, we get:
z=(2cos(t))(2sin(t))=4sin(t)cos(t)
Therefore, the vector function that represents the curve of intersection is:
𝐫(t)=x(t),y(t),z(t)=2cos(t),2sin(t),4sin(t)cos(t)
where t ranges from 0 to 2π.
xleb123

xleb123

Skilled2023-05-10Added 181 answers

Answer:
2cos(t),2sin(t),4sin(t)cos(t)
Explanation:
To represent the curve of intersection between the cylinder x2+y2=4 and the surface z=xy as a vector function, we can parameterize the curve using a parameter t.
Let's consider the equation of the cylinder:
x2+y2=4
We can rewrite this equation in polar coordinates as r=2, where r represents the distance from the origin. Now, we can express x and y in terms of t as x=2cos(t) and y=2sin(t), where t ranges from 0 to 2π.
Substituting these values into the equation of the surface z=xy, we obtain:
z=(2cos(t))(2sin(t))=4sin(t)cos(t)
Hence, the vector function that represents the curve of intersection is:
𝐫(t)=x(t),y(t),z(t)=2cos(t),2sin(t),4sin(t)cos(t)

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