Answered: 2. Find u × v, v × u, and v × v. u =…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Can you show all work and the picture is for #9

2. Find u × v, v × u, and v × v.

u = 5i + 6k
v = 6i + 7j − 6k.

(a) u × v

(b) v × u

(c) v × v

3.Find u × v, v × u, and v × v.

u = <1,-9,6>
v = <-2,3,-7>
(a) u × v

(b) v × u

4.Consider the following.

u = <4,-1,0>
v = <-4,5,0>

Find u ✕ v.

Determine if u ✕ v is orthogonal to both u and v by finding the values below.

u · (u ✕ v) =

v · (u ✕ v) =

u × v is orthogonal to both u and v.

u × v is not orthogonal to both u and v.

5.Consider the following.

u = <-4,3,3>
v = <0,1,7>

Find u ✕ v.

Determine if u ✕ v is orthogonal to both u and v by finding the values below.

u · (u ✕ v)

v · (u ✕ v)

u × v is not orthogonal to both u and v.

u × v is orthogonal to both u and v.

6.Find u × v and show that it is orthogonal to both u and v.

u = <-1,0,7>

v = <8,-4,0>

7.Find a unit vector that is orthogonal to both u and v.

u=<-8,-6,4>

v=<15,-18,-1>

8.Find the area of the triangle with the given vertices.

(Hint: (1/2)||u ✕ v||

is the area of the triangle having u and v as adjacent

sides.)

A(0, 0, 0), B(4, 0, 6), C(−2, 1, 0)

9.The brakes on a bicycle are applied using a downward force of F = 24 pounds on the pedal when the crank makes a 40° angle with the horizontal (see figure). The crank is 6 inches in length. Find the torque at P. (Round your answer to two decimal places.)

10.Find u · (v × w).

u	=	i
v	=	j
w	=	k

11. Find u · (v × w).

u =
<3, 0, 0>

v =
<10, 10, 10>

w =
<0, 3, 3>

12.Find the determinant of the matrix, if it exists.

-5	0
2	-9

13.Find the determinant of the matrix.

1	3	0
0	-2	1
0	3	-3

Determine whether the matrix has an inverse, but don't calculate the inverse.

The matrix has an inverse.

The matrix does not have an inverse.

14. Find the magnitude and direction (in degrees) of the vector, assuming 0 ≤ θ < 360. (Round the direction to two decimal places.)

v =
<6, 8>

|?|

θ= °

15. Find the magnitude and direction (in degrees) of the vector, assuming 0 ≤ θ < 360. (Round the direction to two decimal places.)

? =
<−8, 15>

|?|

θ= °

16. Find the cross product of the unit vectors. Sketch the result.

k × i

17. Determine whether each point lies on the line.

x = −3 + t, y = 3t, z = 4 + t

(a)

(0, 9, 7)

Yes/No

(b)

(3, 4, 7)

Yes/No

(c)

(−5, −6, 2)

Yes/No

18. Find sets of parametric equations and symmetric equations of the line that passes through the given point and is parallel to the given vector or line. (For each line, write the direction numbers as integers.)

Point

Parallel to

(0, 0, 0)

v =

<8, 1, 4>

(a) parametric equations (Enter your answers as a comma-separated list.)

(b) symmetric equations

8x = y = 4z

4x = y = 8z

x/8=y=z/4

x/4=y=z/8

19. Find sets of parametric equations and symmetric equations of the line that passes through the given point and is parallel to the given vector or line. (For each line, write the direction numbers as integers.)

Point		Parallel to
(−3, 0, 3)		v = 8i + 4j − 6k

(a) parametric equations (Enter your answers as a comma-separated list.)

(b) symmetric equations

x/8=y=z/6

(x+3/8)=y/4=(3-z/6)

8x=y/4=6z

(x-3/8)=y=z/6

20.Find sets of parametric equations and symmetric equations of the line that passes through the two points (if possible). (For each line, write the direction numbers as integers.)

(0, 0, 16), (12, 12, 0)

(a) parametric equations (Enter your answer as a comma-separated list of equations in terms of x, y, z, and t.)

(b) symmetric equations

12x = 12y = 16z − 4

3x = 3y = 4z − 16

x/12=y/12=(z-4/16)

x/3=y/3=(z-16/-4)

(x-4/12)=(y-4/12)=(z-4/16)

21.Find a set of parametric equations of the line with the given characteristics. (Enter your answers as a comma-separated list.)

The line passes through the point

(5, 6, 7)

and is parallel to the xz-plane and the yz-plane.

22. Find the coordinates of the point P on the line and a vector v parallel to the line.

x = 7t, y = 8 − t, z = 3 + 2t

P(x, y, z) = ( )

v =

23. Find the coordinates of a point P on the line and a vector v parallel to the line.

(x-3/9)=(y+6/4)=z+7

P(x,y,z)=

24. Determine whether the lines are parallel or identical.

x = 6 − 3t, y = −4 + 4t, z = 5 + 8t

x = 6t, y = 4 − 8t, z = 21 − 16t

25. Determine whether the lines intersect, and if so, find the point of intersection and the cosine of the angle of intersection.

x/3=(y-2/-1)=z+1, (x-1/4)=y+2=(z+3/-3)

26. Find an equation of the plane that passes through the given point and is perpendicular to the given vector or line.

Point		Perpendicular to
(4, 5, −9)		n = j

27. Find an equation of the plane that passes through the given point and is perpendicular to the given vector or line.

Point is (8,2,2)

Perpendicular to (x-1/4)=y+2=(z+8/-8)

28. Find an equation of the plane that passes through the point (8, -6, 4) and contains the line given by the following equation.

x/2=(y-4/-1)=z

29. Determine whether the planes are parallel, orthogonal, or neither.

4x − 2y + z = 4

x + 6y + 8z = 1

Find the angle between the planes.

30. Determine whether the planes are parallel, orthogonal, or neither.

2x − 10y − 2z = 6

3x − 15y − 3z = −5

Find the angle between the planes.

The image depicts a foot pressing down on a bicycle pedal, illustrating the mechanics of applying force to a pedal crank. The diagram includes the following details:

1. **Pedal Crank and Rotation**: The pedal crank has a length of 6 inches, marked from point P, the center of the pedal rotation, to the position where the force is applied.

2. **Force Application (F)**: A foot is shown applying a force (F) on the pedal. This force is directed downward at an angle.

3. **Angle of Force**: The angle between the line of force and a line perpendicular to the crank (at point of contact with the pedal) is labeled as 40 degrees.

This setup illustrates the concept of torque, where the force applied at a distance from the pivot point (P) causes rotational motion in the bicycle mechanism.

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2. Find u × v, v × u, and v × v. u = 5i + 6k v = 6i + 7j − 6k. (a) u × v (b) v × u (c) v × v