MHB Does this triangle have to be flipped when finding the magnitudes and angles?

AI Thread Summary
In relative velocity problems, the direction of the plane does not need to be flipped when drawing the triangle for magnitudes and angles. The example provided illustrates how to calculate the resultant velocity of a plane considering wind direction using both the Law of Cosines and vector addition. The angle between the plane's velocity vector and the wind vector is crucial for accurate calculations. Both methods yield the same resultant velocity of approximately 692.58 km/h. Understanding the correct angles and vector directions is essential for solving these types of problems effectively.
Raerin
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In those kind of relative velocity questions in calculus & vectors class, does the direction of the plane need to be flipped when drawing the triangle to find the magnitudes and angles?

Example of a question:
An airline pilot has her controls set to fly at an air speed of 615 km/h at an azimuth bearing of 40 degrees. A wind is blowing from an azimuth bearing of 205 degrees at 80 km/h. Determine the velocity of the plane relative to the ground.

For this question, I also have troubles finding the angles between vectors when I draw the triangle. A link to an illustration on Paint or some other drawing program would be really helpful!

Thanks :)
 
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Here are two ways to work the problem:

a) Law of Cosines:

Consider the following diagram:

View attachment 1546

The angle between the plane's velocity vector and that of the wind is:

$$\theta=\left(180-((90-(205-180))-(90-40)) \right)^{\circ}=165^{\circ}$$

Hence:

$$R=\sqrt{615^2+80^2-2\cdot615\cdot80\cos\left(165^{\circ} \right)}\approx692.583642102$$

b) Vector addition:

$$\vec{R}=\left\langle 615\cos\left(50^{\circ} \right)+80\cos\left(65^{\circ} \right),\sin\left(50^{\circ} \right)+80\sin\left(65^{\circ} \right) \right\rangle$$

$$R=\left|\vec{R} \right|=\sqrt{\left(615\cos\left(50^{\circ} \right)+80\cos\left(65^{\circ} \right) \right)^2+\left(\sin\left(50^{\circ} \right)+80\sin\left(65^{\circ} \right) \right)^2}\approx692.583642102$$
 

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