Three vectors ModifyingAbove a With right-arrow, ModifyingAbove b With right-arr

Question

Three vectors ModifyingAbove a With right-arrow, ModifyingAbove b With right-arrow, and ModifyingAbove c With right-arrow, each have a magnitude of 43.0 m and lie in an xy plane. Their directions relative to the positive direction of the x axis are 28.0 , 192 , and 310 , respectively. What are (a) the magnitude and (b) the angle of the vector ModifyingAbove a With right-arrow plus ModifyingAbove b With right-arrow plus ModifyingAbove c With right-arrow (relative to the +x direction in the range of (-180°, 180°)), and (c) the magnitude and (d) the angle of ModifyingAbove a With right-arrow minus ModifyingAbove b With right-arrow plus ModifyingAbove c With right-arrow in the range of (-180°, 180°)? What are (e) the magnitude and (f) the angle (in the range of (-180°, 180°)) of a fourth vector ModifyingAbove d With right-arrow such that left-parenthesis ModifyingAbove a With right-arrow plus ModifyingAbove b With right-arrow right-parenthesis minus left-parenthesis ModifyingAbove c With right-arrow plus ModifyingAbove d With right-arrow right-parenthesis equals 0?

Explanation / Answer

Ans:-

Adding vectors is much easier than it seems, but it is very easy to confuse signs, and drop a minus sign or something. But here is the basic principle: take every vector and add their horizontal and vertical components, and then you get a resulting vector.

So, in this case:

(a) & (b):
a + b + c =

[a(h), a(v)] + [b(h), b(v)] + [c(h), c(v)] =

(a(h) + b(h) + c(h), (a(v) + b(v)) + c(v)) =

((43m)cos(28) + (43m)cos(192) + (43m)cos(310), (43m)sin(28) + (43m)sin(192) + (43m)sin(310)) =

((37.97 + -42.06 + 27.64), (20.19+ -8.94 + -32.94)) =

(23.55, -21.69) is your resultant vector in component form. To find the magnitude, use a^2 + b^2 = c^2

23.55^2 + (-21.69)^2 = c^2
c = 32.02 meters

So, to find the angle (opposite over adjacent is tangent)
tan (angle) = -21.69/ 23.55
invtangent (-21.69/ 23.55) = -42.65deg

HOWEVER!!!! Make sure that the angle makes sense. For example, if the result is 45 degrees, is it really 45 degrees? Or could it be 135 degrees. Both angle measures give the same answer. So always draw a picture to make sure that your answer corresponds with where it should be. In our case, -21.69 and 23.55 are both positive, so you know that the result vector is in the fourth quadrant,

with c and d, repeat the process, but check your signs. And with e and f, solve just like a normal equation:
(a + b) - (c + d) = 0
a + b = c + d
a + b - c = d
(a(h) + b(h) - c(h), (a(v) + b(v)) - c(v)) =d

((43m)cos(28) + (43m)cos(192) - (43m)cos(310), (43m)sin(28) + (43m)sin(192) - (43m)sin(310)) = d

((37.97 + -42.06 - 27.64), (20.19+ -8.94 + 32.94)) = d
((-31.73,44.19) = d

is your resultant vector in component form. To find the magnitude, use a^2 + b^2 = d^2
d= 54.40m

angle = tan-1(44.19/-31.73)

angle = -54.32deg with resp to +vex axis

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