Graphical Reasoning

[nycmathguy]

Section 6.3

I am having a hard time forming the needed formula to answer parts (b - d). Can you do part (a)? I will then use your formula to continue on my own. Thank you so much.

 

[MathLover1]

F1=<10,0> and F2=5<cos(theta), sin(theta)>

a.
find ||F1+F2|| as function of theta

First we find the component of F1+F2

F1+F2=<10,0>+5<cos(theta), sin(theta)>
F1+F2=<10,0>+<5cos(theta), 5sin(theta)>
F1+F2=<10+5cos(theta),0+ 5sin(theta)>
F1+F2=<10+5cos(theta), 5sin(theta)>

Now, for magnitude of F1+F2

||F1+F2|sqrt((10+5cos(theta))^2+(5sin(theta))^2 )

||F1+F2|sqrt(100+100cos(theta)+25cos^2(theta)+25sin^2(theta) )
||F1+F2|sqrt(100+100cos(theta)+25(cos^2(theta)+sin^2(theta) ))...using cos^2(theta)+sin^2(theta)=1

||F1+F2||=sqrt(100+100cos(theta)+25)

||F1+F2||=sqrt(125+100cos(theta))

||F1+F2||=sqrt(25(5+4cos(theta)))

||F1+F2||=5sqrt(5+4cos(theta))

b.

Given
F1=<10,0>
F2=5<cos(theta), sin(theta)>
We have to Use a graphing utility to graph the function in part for a. for 0 <=theta <=2pi.
The graphing utility to graph the function is shown below.

c.
form part b range for 0<=θ<2 π
.
||F1+F2||=5sqrt(5+4cos(θ))

From here we get the maximum value of θ at θ = 0 °
||F1+F2||=5sqrt(5+4cos(0))
||F1+F2||=5sqrt(5+4*1)
||F1+F2||=5sqrt(9)
||F1+F2||=5*3
||F1+F2||=15

Now we get the minimum value of θ at θ = pi

||F1+F2||=5sqrt(5+4cos(pi))
||F1+F2||=5sqrt(5+4(-1))
||F1+F2||=5sqrt(5-4)
||F1+F2||=5sqrt(1)
||F1+F2||=5

The range of the function for 5< =θ<15 .

d. The magnitude of a vector is always a positive number or zero it cannot be a negative number. A vector is a quantity described by a magnitude and a direction. The magnitude is always positive or zero. A negative sign in front of a vector indicates the same magnitude but in the opposite direction. The negative sign is part of the direction rather than the magnitude.

 

[nycmathguy]

F1=<10,0> and F2=5<cos(theta), sin(theta)>

a.
find ||F1+F2|| as function of theta

First we find the component of F1+F2

F1+F2=<10,0>+5<cos(theta), sin(theta)>
F1+F2=<10,0>+<5cos(theta), 5sin(theta)>
F1+F2=<10+5cos(theta),0+ 5sin(theta)>
F1+F2=<10+5cos(theta), 5sin(theta)>

Now, for magnitude of F1+F2

||F1+F2|sqrt((10+5cos(theta))^2+(5sin(theta))^2 )

||F1+F2|sqrt(100+100cos(theta)+25cos^2(theta)+25sin^2(theta) )
||F1+F2|sqrt(100+100cos(theta)+25(cos^2(theta)+sin^2(theta) ))...using cos^2(theta)+sin^2(theta)=1

||F1+F2||=sqrt(100+100cos(theta)+25)

||F1+F2||=sqrt(125+100cos(theta))

||F1+F2||=sqrt(25(5+4cos(theta)))

||F1+F2||=5sqrt(5+4cos(theta))

b.

Given
F1=<10,0>
F2=5<cos(theta), sin(theta)>
We have to Use a graphing utility to graph the function in part for a. for 0 <=theta <=2pi.
The graphing utility to graph the function is shown below.

c.
form part b range for 0<=θ<2 π
.
||F1+F2||=5sqrt(5+4cos(θ))

From here we get the maximum value of θ at θ = 0 °
||F1+F2||=5sqrt(5+4cos(0))
||F1+F2||=5sqrt(5+4*1)
||F1+F2||=5sqrt(9)
||F1+F2||=5*3
||F1+F2||=15

Now we get the minimum value of θ at θ = pi

||F1+F2||=5sqrt(5+4cos(pi))
||F1+F2||=5sqrt(5+4(-1))
||F1+F2||=5sqrt(5-4)
||F1+F2||=5sqrt(1)
||F1+F2||=5

The range of the function for 5< =θ<15 .

d. The magnitude of a vector is always a positive number or zero it cannot be a negative number. A vector is a quantity described by a magnitude and a direction. The magnitude is always positive or zero. A negative sign in front of a vector indicates the same magnitude but in the opposite direction. The negative sign is part of the direction rather than the magnitude.

I don't know how I am ever going to repay you for all the math help you have provided since we met. Honestly, this question is too hard for precalculus. Of course, this is only my opinion.

I took precalculus in the Spring 1993 semester at Lehman College. The graduate school student working on his PH.D. allowed to teach MA172 never talked about vectors, navigation, bearing, polar equations, etc. I got an A minus in the course. The toughest concept in that evening class , if memory serves me right, was learning to find the domain and range of functions, including trig functions. I don't even think he covered law of sines and law of cosines.