Question 1206226: Three vectors add together so that the resultant is zero. Vector A points 75.0° north of East. Vector B points due west. Vector C points due south and has magnitude of 185 metres. Find the magnitude of A and B.
Answer by math_tutor2020(3817)   (Show Source):
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| Vector | Magnitude | Angle | | A | r | 75 | | B | s | 180 | | C | 185 | 270 |
Each vector can be written in component form: (x,y) = (m*cos(theta), m*sin(theta))
where,
m = magnitude
theta = angle
vector A: (r*cos(75), r*sin(75))
vector B: (s*cos(180), s*sin(180)) = (-s, 0)
vector C: (185*cos(270), 185*sin(270)) = (0, -185)
In other words,
vector A: (r*cos(75), r*sin(75))
vector B: (-s, 0)
vector C: (0, -185)
Add the vectors by adding the corresponding components.
If (x,y) and (v,w) are two vectors, then they sum to (x+v, y+w).
This can be extended to 3 vectors or more.
A+B+C = (r*cos(75) - s, r*sin(75) - 185)
This resultant vector is stated to be the zero vector (0,0).
Set each component equal to zero so we form this system of equations.
r*cos(75) - s = 0
r*sin(75) - 185 = 0
Solving the second equation for variable r gets us:
r = 185/sin(75) = 191.526093 approximately.
Make sure that your calculator is in degrees mode.
Then,
r*cos(75) - s = 0
s = r*cos(75)
s = 191.526093*cos(75)
s = 49.570601 approximately
In summary we found these approximations,
r = 191.526093
s = 49.570601
which represent the magnitudes of vectors A and B respectively.
Round these decimal values however needed.
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