Wow, tried helping a friend with a vector problem and I can't get it

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In summary, the hiker traveled a total distance of 1550m in a direction 21 degrees north of east, then 41 degrees east of south, and finally 18 degrees south of west, with the resulting triangle forming a closed loop. Two standard approaches to solving this problem involve resolving all vectors to components or using the cosine and sine rules. The mistake in the given attempt was treating the third angle as 18 degrees north of west instead of 180-18 degrees.
  • #1
1MileCrash
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Homework Statement



A hiker starts from the trailhead and hikes 1550m in a direction 21* north of east. Then, using her compass, she hikes the second leg of her journey in a direction 41* east of south. Finally, she hikes an additional distance in the direction 18* north of west. At the end of the third leg of her trip, she is back at the trailhead where she started. What is the distance she hiked along the second leg?

Homework Equations





The Attempt at a Solution



Ok.. so I took this as "all of the vectors sum to 0."

Converting all angles to "actual" angles, I have 21*, 311*, and 108*, respectively.

I then figured that I could make a system of equations, one equation for each coordinate so that they sum to 0. let n be the magnitude of the second vector, and q be the magnitude of the third.

ncos(311) + qcos(108) + 1550cos(21) = 0

and

nsin(108) + qsin(311) + 1550sin(21) = 0

Valid?

I then made everything into polynomials for simplicity and got these two equations:

.95q - .75n = -555
and
-.3q + .65n = -1447.04

Which shows that q = -3684 and that n = -3926.

So, n should be the magnitude of the second vector of the trip, right? My work doesn't lead to an answer choice.
 
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  • #2
There are two standard approaches to this:
draw the resulting triangle and either;
1. resolve all vectors to components (put N along y-axis and E along x for eg); or
2. work out known interior angles and lengths: use the cosine and sine rules.

You know one length and all the angles so method #2 looks good for this one.

You tried #1 - treating "actual" angles as measured anticlockwise from due east.
But I think you want to take a closer look at that third angle 18 degrees north of West is 180-18 isn't it?

Apart from not leading to an answer choice - you should be able to see that the result is not even reasonable.
The second angle goes almost SE and only ends up 18 degrees below the start ... try to get into the habit of reality-checking your results.
Especially for multiple choice - because sneaky examiners will include common mistakes in the available choices.
 
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FAQ: Wow, tried helping a friend with a vector problem and I can't get it

1. How do I approach a vector problem?

When approaching a vector problem, it is important to first understand the basic principles of vectors, such as magnitude and direction. Then, carefully read the problem and identify what is given and what is being asked for. Finally, use mathematical equations and diagrams to solve the problem.

2. What are some common mistakes people make when solving vector problems?

Some common mistakes people make when solving vector problems include not clearly labeling the vectors, using incorrect equations, and not considering the direction of the vectors. It is also important to double-check calculations and units to ensure accuracy.

3. How do I know which mathematical equations to use for a vector problem?

The mathematical equations used for solving vector problems depend on what is given and what is being asked for. For example, if the problem involves finding the resultant vector, you would use the Pythagorean theorem and trigonometric functions. If the problem involves finding the projection of one vector onto another, you would use dot products and unit vectors.

4. Can you provide an example of solving a vector problem?

Sure! Let's say you are given two vectors, A and B, with magnitudes 5 and 3, respectively. Vector A is at an angle of 30 degrees from the horizontal axis, and vector B is at an angle of 60 degrees from the horizontal axis. To find the resultant vector, you would first use the Pythagorean theorem to find the magnitude of the resultant vector: √(5² + 3²) = √34. Then, you would use trigonometric functions to find the direction of the resultant vector: tan^-1(3/5) = 31.8 degrees. Therefore, the resultant vector has a magnitude of √34 and is at an angle of 31.8 degrees from the horizontal axis.

5. What resources can I use to improve my skills in solving vector problems?

There are many resources available to help improve your skills in solving vector problems. Some options include online tutorials, textbooks, practice problems, and seeking help from a teacher or tutor. It can also be helpful to join a study group or work on problems with a friend to discuss and learn from each other's approaches.

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