Farmer wants to Fence off his Four-Sided Plot with missing Side| Vector Addition and Subtraction| Analytical Method| College Physics| Problem 3.22

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?

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Flying in a Straight Line with Rotated Axes| Vector Addition and Subtraction| Analytical Method| College Physics| Problem 3.21

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°

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College Physics 3.18 – Driving in a straight line vs east and north distances


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:

North and East Components of the given displacement
North and East Components of the given displacement

The east distance is the component in the horizontal direction.

D_E=7.50\:km\:\cdot sin\:\left(15^{\circ} \right)

D_E=1.94\:km

The north distance is the vertical component

D_E=7.50\:km\cdot cos\left(15^{\circ} \right)

D_E=7.24\:km

Part B

3.19

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


College Physics 3.17 – Reverse addition of vectors


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.

Reverse addition of the two vectors A and B.
Reverse addition of the two vectors A and B.

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

R=\sqrt{A^2+B^2}

R=\sqrt{\left(18\:m\right)^2+\left(25\:m\right)^2}

R=\sqrt{324+625}

R=\sqrt{949}

R=30.8\:m

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 

tan\:\phi =\frac{A}{B}

tan\:\phi =\frac{18\:m}{25\:m}

\phi =tan^{-1}\left(\frac{18}{25}\right)

\phi =35.75^{\circ}

Therefore, the compass angle is 35.75^{\circ} \:West\:of\:North