A farmer wants to fence off his four-sided plot of flat land. He measures the first three sides, shown as A, B, and C in the figure, and then correctly calculates the length and orientation of the fourth side D. What is his result?

SOLUTION:

We know that the sum of the 4 vectors is zero. So, Vector D is calculated as

We solve for the x-component first. The x-component of vector D is

We solve for the y-component.

Since we already know the x and y component of vector D, we can finally solve for the distance of vector D.

You fly 32.0 km in a straight line in still air in the direction 35.0° south of west.

(a) Find the distances you would have to fly straight south and then straight west to arrive at the same point. (This determination is equivalent to finding the components of the displacement along the south and west directions.)

(b) Find the distances you would have to fly first in a direction 45.0º south of west and then in a direction 45.0° west of north. These are the components of the displacement along a different set of axes—one rotated 45.0°.

Solution:

Part A

The component along the south direction is

The component along the west direction is

Part B

Consider the following figure with the rotated axes x’-y’.

A new landowner has a triangular piece of flat land she wishes to fence. Starting at the west corner, she measures the first side to be 80.0 m long and the next to be 105 m. These sides are represented as displacement vectors A from B in Figure 3.61. She then correctly calculates the length and orientation of the third side C. What is her result?

Solution:

Since the figure is closed, we know that the sum of the three vectors is equal to 0. So, we have

Solving for C, we have

Next, we solve for the x and y components of Vectors A and B.

You drive 7.50 km in a straight line in a direction 15° east of north.

(a) Find the distances you would have to drive straight east and then straight north to arrive at the same point. (This determination is equivalent to find the components of the displacement along the east and north directions.)

(b) Show that you still arrive at the same point if the east and north legs are reversed in order.

Solution:

Part A

Consider the following figure:

The east distance is the component in the horizontal direction.

The north distance is the vertical component

Part B

Based from the figure, we can easily see that the order is reversible in the addition of vectors. We say that

Repeat Problem 3.16 using analytical techniques, but reverse the order of the two legs of the walk and show that you get the same final result. (This problem shows that adding them in reverse order gives the same result—that is, B+A=A+B) Discuss how taking another path to reach the same point might help to overcome an obstacle blocking your other path.

Solution:

Consider the right triangle formed by the legs A, B, and R. We know that A is 18 m, B is 25 m, and we are solving for the magnitude of R. We can do this by using the Pythagorean Theorem. That is

So, the distance is about 30.8 meters from the starting point. To solve for the value of the unknown angle, φ, we can use the tangent function. That is

Solve the following problem using analytical techniques: Suppose you walk 18.0 m straight west and then 25.0 m straight north. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements A and B , as in Figure 3.60, then this problem asks you to find their sum R = A + B .)

Note that you can also solve this graphically. Discuss why the analytical technique for solving this problem is potentially more accurate than the graphical technique.

Solution:

Consider the right triangle formed by the legs A, B, and R. We know that A is 18 m, B is 25 m, and we are solving for the magnitude of R. We can do this by using the Pythagorean Theorem. That is

So, the distance is about 30.8 meters from the starting point. To solve for the value of the unknown angle, θ, we can use the tangent function. That is

So, the compass reading, can be solved by taking the complimentary angle, as shown in the figure.

(b) the magnitude and direction of the displacement from start to finish.In this part of the problem, explicitly show how you follow the steps of the analytical method of vector addition.

Solution:

Part A

The total distance traveled following Path C is

Part B

Treat all motions upward and to the right positive, while downward and to the left negative.

Suppose you first walk 12.0 m in a direction 20º west of north and then 20.0 m in a direction 40.0º south of west. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements A and B , as in Figure 3.56, then this problem finds their sum R = A + B .)