   Chapter 13.2, Problem 27E

Chapter
Section
Textbook Problem

# Find a vector equation for the tangent line to the curve of intersection of the cylinders x2 + y2 = 25 and y2 + z2 = 20 at the point (3, 4, 2).

To determine

To find: A vector equation for the tangent line to the curve of intersection of the cylinders x2+y2=25 and y2+z2=20 at the point (3,4,2).

Explanation

Formula used:

Write the expression to find the vector equation for the tangent line through the point (x0,y0,z0) and parallel to the vector v=a,b,c.

r(t)tangent line=(x0+at)i+(y0+bt)j+(z0+ct)k (1)

The required tangent line passes through the point (3,4,2) and it is parallel to the tangent vector of the curve.

The tangent vector of the curve is the derivative of the vector function of curve.

Calculation of vector function of the curve r(t):

As the curve C passes through the point of inter section of cylinders x2+y2=25 and y2+z2=20 at the point (3,4,2), the projection of curve C onto the xy-plane is contained in the circle x2+y2=25,z=0 and on the cylinder y2+z2=20,z0 near to the point (3,4,2).

Write the first cylinder equation as follows.

x2+y2=25

Consider x- and y-components of the vector function r(t) such that the considered components should satisfy the cylinder equation x2+y2=25 and consist of scalar parameter t.

x=5costy=5sint

The considered x- and y-components are satisfied the cylinder equation x2+y2=25.

Write the second cylinder equation as follows.

y2+z2=20

Rewrite the expression as follows.

z=20y2

Substitute 5sint for y,

z=20(5sint)2=2025sin2t

From the analysis, the vector function r(t) is written as follows.

r(t)=5cost,5sint,2025sin2t

Calculation of tangent vector r(t):

To find the derivative of the vector function, differentiate each component of the vector function.

Differentiate each component of the vector function r(t)=5cost,5sint,2025sin2t as follows.

ddt[r(t)]=ddt(5cost),ddt(5sint),ddt(2025sin2t)

Use the following formula to compute the expression.

ddt[f(t)]=12f(t)ddt[f(t)]ddtcost=sintddtsint=costddtsin2t=2sintcost

Compute the expression ddt[r(t)]=ddt(5cost),ddt(5sint),ddt(2025sin2t) by using the formulae as follows

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