8 Problems Vectors: Components of a vector, resultant vector

(See attached file for full problem description with diagrams)

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Please solve each and give a complete detailed step by step solution to solving each problem no matter how easy. I am trying to learn and not an expert. Give answer and way to solve. Adult student trying to make my way through all this and need your help!

1) Note: You are not given the direction moved after any of the 90 degree turns, so there could be more than one answer.
A shopper pushing a cart through a store moves 30.0m south down onee aisle, then makes a 90 degree turn and moves 19.0m. He then makes another 90 degree turn and moves 14.0m.
a) What is the magnitude of the smallest possible displacement the shopper could have? Answer in m.
b) At how many degrees from due south is this displacement. Answer in degrees.
c) What is the magnitude of the largest possible displacement the shopper could have? Answer in m.
d) At how many degrees from due south is this displacement? Answer in degrees.

2) Consider the two vectors M = (a,b) = ai(hat) + bj(hat) and N=(c,d) = ci(hat) + dj(hat), where a=4, b=4,c=-2, and d=2. a and c represent the x displacement and b and d represent the y displacement in a Cartesian xy coordinate system. Note: i hat and j hat represent unit vectors (i.e. vectors of length 1.) in the x and y directions, respectively.
a) What is the value of the scalar product M dot M?
1) 2ab
2) a^2-2ab+b^2
3) SQRT(a^2+b^2)
4) a^2+b^2
5) a^2+2ab+b^2
6) a-b
7) a+b
8) -2ab
9) a^2-b^2

b) What is the value of the scalar product M(hat) dot N(hat)?
1) ad+bc
2) a^2+b^2+c^2+d^2
3) ac-bd
4) ab+cd
5) ab-cd
6) ad-bc
7) ac+bd
8) SQRT(a^2+b^2) + SQRT(c^2+d^2)
9) abcd

3) 12) Consider the two vectors M = (a,b) = ai(hat)+bj(hat) and N=(c,d)=ci(hat)+dj(hat), where a=4,b=4,c=1,d=-1. a and c represent the x displacement and b and d represent the y displacement in a Cartesian xy coordinate system. Note: i hat and j hat represent unit vectors (i.e. vectors of length 1.) in the x and y directions, respectively.
a) What is the value of the scalar product N dot N?
b) What is the value of the scalar product M dot N?

4) All angles are measured in a counterclockwise direction from the positive x axis. A hiker makes 3 straight line walks(A,B and C) in random directions and lengths starting at position (41km,41km), listed below and shown below in the plot.
A 11km at 319 degrees
B 27km at 287 degrees
C 19km at 179 degrees
Scale: 10km =
Select the vector which will return the hiker to the starting point by identifying the vector D (described below) with the diagram above.

1) || D || = 46.363km, θd = 36.9435 degrees
2) || D || = 35.7192km, θd = 5.72855 degrees
3) || D || = 80.7451km, θd = 197.559 degrees
4) || D || = 45.6165km, θd = 153.28 degrees
5) || D || = 59.9282km, θd = 121.511 degrees
6) || D || = 29.9547km, θd = 170.859 degrees
7) || D || = 32.825km, θd = 85.1046 degrees
8) || D || = 34.3031km, θd = 270.108 degrees
9) || D || = 47.7921km, θd = 100.87 degrees
10) || D || = 37.8193km, θd = 312.289 degrees

5) An ant starts at one edge of a long strip of paper that is 36.7 cm wide. She travels at 2.2 cm/s at an angle of 46 degrees with the long edge. How long will it take her to get across? Answer in units of seconds?

6) As it passes over an island, the eye of a hurricane is moving in a direction 76 degrees north of west with a speed of 57 km/h. Three hours later, it shifts due north, and its speed slows to 26km/h. How far from the island is the eye 4.50 hours after it passes over the island? Answer in km.

7) Vector B has x,y,andz components of 5.8,2.6, and 9.5 units respectively.
a) Calculate the magnitude of B.
b) What is the angle between B and the x axis? Answer in degrees.

8) Consider the vectors A (OA = xai(hat) + yaj(hat)) and B( OB = xbi(hat) + ybj(hat)) shown in the figure, not drawn to scale, where OA isin the first quadrant and OB is in the second quadrant. Let the vector C (OC) be the sum of A and B, that is, C=A+B.
Given:
||A|| = |OA| = 5.86
||B|| = |OB| = 1.96
angle OA = theta1 = 25.6 degrees
angle OB = theta2 = 104 degrees and
angle OC = theta c.

a) Find the x component of C.
b) Find the y component of C
c) Let theta be the angle measured from the postive x direction to the vector C. Identify the correct expression below.

1) tan theta c = ya -yb / xa - xb
2) tan theta c = yb -ya / xa - xb
3) tan theta c = xa + xb / ya +yb

4) tan theta c = ya +yb / xa + xb

5) tan theta c = (ya / xa) + (yb / xb)

6) tan theta c = (xa /ya) + ( xb / y b)
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