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Study23
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Stuck for a long time :(..........Please help me with vector addition! Illustration below. I need to find the direction and magnitude of the sum vector. The correct answer is 41.62, 42 degrees from x to y
 10 months ago
 10 months ago
Study23 Group Title
Stuck for a long time :(..........Please help me with vector addition! Illustration below. I need to find the direction and magnitude of the sum vector. The correct answer is 41.62, 42 degrees from x to y
 10 months ago
 10 months ago

This Question is Closed

Study23 Group TitleBest ResponseYou've already chosen the best response.1
dw:1381109606282:dw (Construct c = a + b)
 10 months ago

Study23 Group TitleBest ResponseYou've already chosen the best response.1
dw:1381109717758:dw
 10 months ago

wolfe8 Group TitleBest ResponseYou've already chosen the best response.1
I believe you are supposed to add them like this: dw:1381110194843:dw
 10 months ago

Study23 Group TitleBest ResponseYou've already chosen the best response.1
I was trying to use the component method (x and y components)
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
What are the magnitudes of the two vectors?
 10 months ago

Study23 Group TitleBest ResponseYou've already chosen the best response.1
Oh! I forgot to write that! One second...
 10 months ago

Study23 Group TitleBest ResponseYou've already chosen the best response.1
dw:1381111020150:dw
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
dw:1381111017373:dw
 10 months ago

Study23 Group TitleBest ResponseYou've already chosen the best response.1
255 or 75?
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
\(R_x = A_x + B_x = 5 \cos20 + 5 \cos255 \) \(R_y = A_y + B_y = 5 \sin20 + 5 \sin255 \) \(R = \sqrt{R_x^2 + R_y^2} \) \(\theta = \tan^{1} \dfrac{R_y}{R_x} \)
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
An angle of 75 deg down form the negative yaxis is an angle of 255 deg from the positive xaxis. The advantage of using the 255 deg angle is that the sin and cos will automatically be negative numbers since it's in the third quadrant.
 10 months ago

Study23 Group TitleBest ResponseYou've already chosen the best response.1
Oh!! Okay That makes sense!
 10 months ago

Study23 Group TitleBest ResponseYou've already chosen the best response.1
@mathstudent55 do you mind If I calculate that out to see if I get the correct answer? I've been half an hour, so please don't leave :)
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
No problem.
 10 months ago

Study23 Group TitleBest ResponseYou've already chosen the best response.1
Hmmm I get 4.61 for magnitude?
 10 months ago

Study23 Group TitleBest ResponseYou've already chosen the best response.1
7 degrees?
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
I got 4.62 magnitude, and 42.5 deg.
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
What did you get for Rx and Ry?
 10 months ago

Study23 Group TitleBest ResponseYou've already chosen the best response.1
Rx: \(\ 5cos20+5cos255 \) \(\ = \) \(\ 0.13... \) Rx: \(\ 5sin20 + 5sin255 \) \(\ = \) \(\ 0.416 \)
 10 months ago

Study23 Group TitleBest ResponseYou've already chosen the best response.1
\(\ \huge \text{The latter should be } Ry. \)
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
This is what I get for the components of the resultant. \(R_x = 5 \cos 20 + 5 \cos 255 = 4.6985 + (1.29409) = 3.4044 \) \(R_y = 5 \sin 20 + 5 \sin 255 = 1.7101 + (4.82963) = 3.11952 \)
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
\( R^2 = R_x^2 +R_y^2 \) \( R = \sqrt{(3.4044)^2 + (3.11952)^2} \) \( R = 4.6175 \) \( \theta = \tan^{1}\dfrac{R_y}{R_x} \) \( \theta = \tan^{1} \dfrac{3.11952}{3.4044} \) \(\theta = 42.5^o\)
 10 months ago

Study23 Group TitleBest ResponseYou've already chosen the best response.1
Hmmm So perhaps the answer key is incorrect?
 10 months ago

Study23 Group TitleBest ResponseYou've already chosen the best response.1
I hope this not too much, but there is one other problem I am stuck on... this time with vector subtraction. Do you mind helping me with that?
 10 months ago

Study23 Group TitleBest ResponseYou've already chosen the best response.1
dw:1381113129371:dw
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
a  b = a + (b)
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
The new vector is b. Now find x and y components of vector a and vector b. Add the components together and find the magnitude of the resultant. Then use the inverse tangent to find the angle of the resultant. dw:1381113881005:dw
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
dw:1381114004562:dw
 10 months ago

Study23 Group TitleBest ResponseYou've already chosen the best response.1
Okay, and I would still add the components except add the OPPOSITE of the b components?
 10 months ago

Study23 Group TitleBest ResponseYou've already chosen the best response.1
would x still be cos and y still be sin in this case? I always forget when those are switched
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
\(A_x = 8 \cos 135\) \(A_y = 8 \sin 135 \) \(B_x = 4.5 \cos 240 \) \( B_y = 4.5 \sin 240 \) \(R_x = A_x + B_x = 8 \cos 135 + 4.5 \cos 240 = 7.90685\) \(R_y = A_y + B_y = 8 \sin 135 + 4.5 \sin 240 = 1.75974\)
 10 months ago

Study23 Group TitleBest ResponseYou've already chosen the best response.1
Where do the 135 and 240 come from?
 10 months ago

Study23 Group TitleBest ResponseYou've already chosen the best response.1
It's subtraction, so shouldn't it be  4.5cos240?
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
Let me explain that again.
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
dw:1381114977171:dw
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
These are the original vectors A and B, ok?
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
dw:1381115051999:dw
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
Instead of A + B, they want A  B, right?
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
Mathematically speaking, A + B is the same as A + (B). For example, 4  3 = 4 + (3)
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
So in order to perform the subtraction of vectors A  B, we can instead add vectors A and B.
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
The first step now is to find what the vector B is.
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
If vector \(B = B_x + B_y\), then \(B = ( B_x + B_y) =  B_x + ( B_y)\)
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
If B is the vector in the figure below, dw:1381115308130:dw
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
dw:1381115353688:dw
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
Those are the components of B. Then this is B: dw:1381115399994:dw
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
Since vector B goes up to the right, vector B goes down to the left.
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
Now remember that we need to add vectors A and B to subtract A  B.
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
dw:1381115515926:dw
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
There you have vector A and vector B. Now we need to add them.
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
First, let's translate the angles of the vectors to angles starting at the positive x axis.
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
dw:1381115630429:dw
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
Now we need to add a vector with magnitude 8 at 135 degrees and a vector of magnitude 4.5 at 240 degrees.
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
Now we get x and y components of both vectors.
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
\(A_x = 8 \cos 135 = 5.65685\) \(A_y = 8 \sin 135 = 5.65685\) \(B_x = 4.5 \cos 240 = 2.25\) \(B_y = 4.5 \sin 240 = 3.89711 \)
 10 months ago

Study23 Group TitleBest ResponseYou've already chosen the best response.1
Ahhh! That makes more sense!
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
Now to find the resultant we add the x components to find the x component of the resultant, and we do the same for the y components.
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
\( R_x = A_x + B_x = 5.65685 + (2.25) = 7.90685 \) \(R_y = A_y + B_y = 5.65685 + (3.89711) = 1.75974 \)
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
Now that we have the x and y components of the resultant, we can use the Pythagorean theorem to find the magnitude of the resultant. \(R = \sqrt{R_x^2 + R_y^2} \) \(R = 8.1\) We find the angle by using the inverse tangent: \(\theta = \tan^{1} \dfrac{R_y}{R_x} \) \( \theta = \tan^{1} \dfrac{1.75974}{7.90685} \) \(\theta = 12.5^o\)
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
dw:1381116568714:dw
 10 months ago

Study23 Group TitleBest ResponseYou've already chosen the best response.1
Thank you so much @mathstudent55! You're help is invaluable! I was stuck on these for soooo long, thank you thank you!
 10 months ago

mathstudent55 Group TitleBest ResponseYou've already chosen the best response.1
You're welcome. BTW, notice how I drew the direction in the last picture. The direction is 12.5 deg up from the negative xaxis, or 167.5 deg measured counterclockwise from the positive xaxis. The answer is 8.1 at 167.5 deg.
 10 months ago
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