In the vector sum A + B = C , vector A has a magnitude of 13.1 m and is angled 49° counterclockwise from the +x direction, and vector C has a magnitude of 15.0 m and is angled 20.0° counterclockwise from the -x direction. What are
(a) the magnitude and
(b) the direction (relative to +x of vector B?
[[[Warning! Make a graphical solution first so you will know what quadrant B lies in.]]]
magnitude = _____ m
direction = _____ degrees
.. i cant figure out how to exactly find out where B is located!
OMG. i figured it out! thanks for lookin' tho. ;)
~lor
A + B = C.
13.1m[49o] + B = 15m[200o],
8.59+9.89i + B = -14.1+(-5.13i),
a. B = -22.69-15.02i = 27.21m[33.5o] S. of W.
b. Direction = 33.5o S of W. = 213.5o CCW.
To find the magnitude and direction of vector B, we can use vector addition and some trigonometry.
(a) To find the magnitude of vector B, we can use the magnitude and direction of vectors A and C. By drawing a graphical representation of the vectors, you can visualize that vector B completes a triangle with vectors A and C.
Using the law of cosines:
C^2 = A^2 + B^2 - 2AB*cos(theta)
where C is the magnitude of vector C, A is the magnitude of vector A, B is the magnitude of vector B, and theta is the angle between vectors A and B.
Rearranging the formula, we can solve for B:
B^2 = C^2 + A^2 - 2AC*cos(theta)
Plugging in the given values:
B^2 = (15.0 m)^2 + (13.1 m)^2 - 2 * (15.0 m) * (13.1 m) * cos(theta)
Once you calculate B^2, take the square root to find the magnitude of vector B.
(b) To find the direction of vector B, we need to determine the angle it makes with the +x direction. To do this, we can use the law of sines:
sin(alpha) / A = sin(theta) / B
where alpha is the angle between the positive x-axis and vector A, and theta is the angle between vectors A and B.
Rearranging the formula, we can solve for alpha:
sin(alpha) = (sin(theta) / B) * A
Plugging in the known values:
sin(alpha) = (sin(49°) / B) * 13.1 m
Solve for alpha using inverse sine (arcsin) to find the angle between the +x direction and vector A.
Finally, to find the direction of vector B, subtract the angle alpha from the angle between the +x direction and vector C (20°). If the resulting angle is in the counterclockwise direction from the +x direction, then the direction of vector B will be positive. If it is in the clockwise direction, then the direction of vector B will be negative.
By following these steps, you should be able to find both the magnitude and direction of vector B.