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2 - 1.8 The Component Method of Vector Addition Many times...

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1.8 The Component Method of Vector Addition Many times using Law of Sines or Law of Cosines will work when doing Many times, using Law of Sines or Law of Cosines will work when doing vector addition. However, there are times when the geometry can get complicated, and a better way presents itself. This is the component method of vector addition . It works every time and for any number of vectors. ĺ ĺ Let s say I have two vectors A and B , and I know the resultant vector C , such that C = A + B . ĺ ĺ ĺ ĺ L t th t i th l d th l k lik th f ll i Let s say the vectors are in the xy-plane and they look like the following: y B ĺ C ĺ x A ĺ
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y Let s go ahead and break A and B down into their x- and y-components. ĺ ĺ A ĺ B ĺ C ĺ B y ĺ B ĺ Now let s add the x- and y-components of vector C to the plot ĺ C y ĺ x A x ĺ A y ĺ x vector to the plot. C x ĺ ĺ ĺ ĺ ĺ ĺ ĺ Now it s easy to see that A x + B x = C x and A y + B y = C y y y C ĺ B ĺ C ĺ A y ĺ y C y ĺ x A x ĺ B x ĺ C x ĺ x
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Once you have the two components of vector C , then how do you calculate its magnitude??? ĺ U th P th Th !!! Use the Pythagorean Theorem!!! C 2 = C x 2 + C y 2 Remember, this method works for any number of vectors, so in general, given n vectors, the scalar components of the resultant vector R x and R y will be: R x = A x + B x +C x + D x + · · · + n x R = A + B +C + D + + n R y = A y + B y +C y + D y + · · · + n y Once the scalar components R and R are calculated then the magnitude Once the scalar components R x and R y are calculated, then the magnitude of R is found by Pythagorean Theorem. R 2 = R x 2 + R y 2
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Example: Staggering out of Fred s at 2 AM on a Friday night, you can t remember where you parked your car. From the bar door you walk 30 m due north, stop, and walk 10 m due east. Here you stop again and then walk 15 m at 30 o south of east. Finally you reach your car. What is the magnitude and direction of the resultant vector (called your displacement vector)? Give the direction relative to east. y, north B = 10 m 30 o The resultant vector, R , would then be: Notice that the following vector equation would be correct: C A = 30 m R R = A + B + C ĺ ĺ ĺ ĺ Car x, east + + So how do I find the magnitude of R ??? ĺ Fred s of We can use the component method of vector addition!!!
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y, north B = 10 m Let s begin by breaking down the vectors A , B , and C into their x- and t C x 30 o y-components.
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