Sail Boat Race Trigonometry, just stuck on first part

[Juntao]

I've added a picture that's part of the problem.

A sailboat race course consists of four legs defined by the displacement vectors A, B, C and D shown above.
The values of the angles are È1 = 420, È2 = 410, and È3 = 270.

The magnitudes of the first three vectors are A = 3.7 km, B = 5.3 km and C = 4.8 km. The finish line of the course coincides with the starting line.

The coordinate system for this problem has positive x to the right, positive y as up and counter-clockwise to be a positive angle.

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Now first thing I should do is break up each vector into its components, and then add up all the x and y components to get vector D.

This is what I've tried.

Vector a
x-component = 3.7*cos 42=2.75km
y-component = 3.7*cos42= 2.48km

vector b
x component = 5.3*cos 41=4.00km
y component = 5.3*sin41= 3.48km

vector c
x component = 4.8*cos 27=4.27km
y component= 4.8*sin27= 2.18km
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vctor d
x component = 11.02km
y component = 8.14km

So for vector d, just add up the first 3 components in their respective columns. However, when I calculate D's distance => sqrt(11.02^2+8.14^2), I get the incorrect answer of 13.70km.
So of course, if this answer is wrong, then most likely I figured out one of the components wrong, but I don't know which ones. Please help.

 

[HallsofIvy]

Did you notice that vector B is going back to the LEFT? Its x component is negative. Both the x component and y component of C will be negative. Also notice that once you have added the components of A, B, and C, the result will be the NEGATIVE of D.

 

[M98Ranger]

It looks like you have correctly calculated the x and y components for vectors A, B, and C. However, when adding them together to find the x and y components for vector D, it seems that you have added the y components instead of subtracting them.

The correct way to find the x and y components for vector D would be:

x component = 2.75 km + 4.00 km - 4.27 km = 2.48 km
y component = 2.48 km + 3.48 km - 2.18 km = 3.78 km

Then, using the Pythagorean theorem to find the magnitude of vector D, we get:

D = sqrt(2.48^2 + 3.78^2) = 4.53 km

So the correct answer for the distance of vector D would be 4.53 km, not 13.70 km.

I hope this helps clarify the mistake and helps you solve the rest of the problem. Remember to always double check your calculations and make sure you are adding and subtracting the correct components. Good luck!