For problems 7 and 8, find the magnitude and direction angle of the given vector.
7. <3, 4>
A magnitude: 5; direction angle: 53.13°
B magnitude: √ 5; direction angle: 53.13°
C magnitude: √5; direction angle: 3.40°
D magnitude: √5; direction angle: 0.80°
E magnitude:5; direction angle: 3.40°
F magnitude: 5; direction angle: 0.80°
8. -3i - 5j
A magnitude: √34; direction angle: 120.96°
B magnitude: √34; direction angle: 239.04°
C magnitude: √34; direction angle: 59.04°
D magnitude: 34; direction angle: 84.94°
E magnitude: 34; direction angle: 95.06°
F magnitude: 34; direction angle: 275.06°
For problems 9 - 11, find the dot product if u = <-1, 3>, v = <2, 4>, and w = <2, -5>.
9. u • v
A 2
B 8
C -10
D 10
E 4
F -3
10. u • w
A -11
B 0
C 1
D -17
E 11
F -1
11. (u + v) • w
A 37
B -37
C -33
D -29
E 29
F 9
For problems 12 and 13, using the theorem given in this lesson, find the angle between the given vectors.
12. u - -<2, 3> and v - <-2, 5>
A 88.3°
B 55.5°
C 11°
D 74.92°
E 1.7°
F 62.4°
13. u - <2, 1> and v - <-1, -3>
A 120°
B 60°
C 315°
D 225°
E 135°
F 45°
14. An airplane is flying on a bearing of 335° at 530 miles per hour. Find the component form of the velocity of the airplane.
A v ≈ <-480.34, -223.99>
B v ≈ <-480.34, 223.99>
C v ≈ <-233.99, -480.34>
D v ≈ <-233.99, 480.34>
E v ≈ <480.34, 223.99>
F v ≈ <233.99, 480.34>
15. An airplane is flying on a bearing of 170° at 460 miles per hour. Find the component form of the velocity of the airplane.
A v ≈ <453.01, 79.88>
B v ≈ <-79.88, 453.01>
C v ≈ <453.01, -79.88>
D v ≈ <-79.88, -453.01>
E v ≈ <79.88, -453.01>
F v ≈ <-453.01, 79.88>
16. Now, assume that the airplane from problem 12 is flying in a wind that is blowing with the bearing 200° at 80 miles per hour. Find the actual ground speed of the airplane.
A 530.79 miles per hour
B 52.52 miles per hour
C 528.19 miles per hour
D 24.09 miles per hour
E 23.08 miles per hour
F 453.01 miles per hour
17. Use the information from problem 13 to find the actual direction (angle) of the airplane. (This is the angle from the horizontal x-axis, not the bearing.)
A 174.32°
B -75.18°
C 250°
D -80°
E 80°
F 84.32°
18. A basketball is shot at a 70° angle with the horizontal with an initial velocity of 10 meters per second. Find the component form of the initial velocity.
A v ≈ <9.40, 3.42>
B v ≈ <-3.42, 9.40>
C v ≈ <3.42, -9.40>
D v ≈ <3.42, 9.40>
E v ≈ <-9.40, -3.42>
F v ≈ <9.40, -3.42>
19. A force of 50 pounds acts on an object at an angle of 45°. A second force of 75 pounds acts on the object at an angle of -30°. Find the direction and magnitude of the resultant force.
A magnitude: 2.14 lbs; direction: -2.14°
B magnitude: 125 lbs; direction: -30°
C magnitude: 125 lbs; direction: -1.22°
D magnitude: 100.33 lbs; direction: 15°
E magnitude: 100.33 lbs; direction: -1.22°
F magnitude: 2.14 lbs; direction: 15°
20. Juana and Diego Gonzales, ages six and four respectively, own a strong and stubborn puppy named Corporal. It is so hard to take Corporal for a walk that they devise a scheme to use two leashes. If Juana pulls with a force of 23 lbs at an angle of 18° and Diego pulls with a force of 27 lbs at an angle of -15°, how hard is Corporal pulling if the puppy holds the children at a standstill?
A 50 lbs
B 4 lbs
C 3 lbs
D 47.95 lbs
E 33 lbs
F 7 lbs
OW! Homework dump!
You'll find help is much more forthcoming if you post one or two problems of the type that are troubling you. I know *I* don't want to play 20 Questions!
20. D
To find the magnitude and direction angle of a vector, you can use the Pythagorean theorem and trigonometry.
For problem 7, the given vector is <3, 4>.
To find the magnitude, use the formula: magnitude = sqrt(x^2 + y^2), where x and y are the components of the vector. In this case, x = 3 and y = 4.
So, magnitude = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5.
To find the direction angle, use the formula: direction angle = arctan(y/x), where arctan is the inverse tangent function. In this case, x = 3 and y = 4.
So, direction angle = arctan(4/3) = 53.13°.
Therefore, the correct answer for problem 7 is: A magnitude: 5; direction angle: 53.13°.
For each subsequent problem, apply the same process to find the correct answer.