   Chapter 13.2, Problem 50E

Chapter
Section
Textbook Problem

# If r(t) = u(t) × v(t), where u and v are the vector functions in Exercise 49, find r'(2).

To determine

To find: The vector r(2) .

Explanation

Given:

r(t)=u(t)×v(t) , u(2)=1,2,1 , u(2)=3,0,4 , and v(t)=t,t2,t3 .

Formula used:

Write the expression to find the derivative of cross product of the two vectors u(t) and v(t) .

ddt[u(t)×v(t)]=u(t)×v(t)+u(t)×v(t) (1)

The vector r(2) is the derivative of the vector function r(t)=u(t)×v(t) at t=2 .

Write the vector function r(t) as follows.

r(t)=u(t)×v(t)

Differentiate on both sides of the expression with respect to t .

ddt[r(t)]=ddt[u(t)×v(t)]

Rewrite the expression as follows.

r(t)=ddt[u(t)×v(t)]

Substitute 2 for t,

r(2)=ddt[u(2)×v(2)] (2)

Use the formula in equation (1) and rewrite the expression in equation (2) as follows.

r(2)=u(2)×v(2)+u(2)×v(2) (3)

Calculation of vector v(2) :

Substitute 2 for t in the expression v(t)=t,t2,t3 ,

v(2)=2,22,23=2,4,8

Calculation of vector v(t) :

The vector v(t) is the derivative of the vector function v(t)=t,t2,t3 .

Differentiate each component of the vector function v(t)=t,t2,t3 to obtain the vector v(t) as follows.

ddt[v(t)]=ddtt,t2,t3

v(t)=ddt(t),ddt(t2),ddt(t3)

Use the following formula and compute the expression.

ddttn=ntn1

Compute the expression v(t)=ddt(t),ddt(t2),ddt(t3) as follows.

v(t)=1,2t,3t2

Calculation of vector v(2) :

Substitute 2 for t in the expression,

v(2)=1,2(2),3(22)=1,4,12

Calculation u(2)×v(2) :

Substitute 3,0,4 for u(2) and 2,4,8 for v(2) in the expression u(2)×v(2) ,

u(2)×v(2)=3,0,4×2,4,8

Rewrite and compute the expression as follows

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