Problem: What are x and y the components of the vector obtained when we add vector A, with magnitude 6 and standard angle 275, to the vector B whose magnitude and standard angle are 5 and 150? What are the magnitude and angle of this resultant vector?
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Solution: The x and y components of A are easily found to be 6 cos( 275 deg) = .5130 and 6 sin( 275 deg) = -5.977; the components of B are similarly found to be -4.327 and 2.504. The x component of the resultant is simply the sum .5130 + -4.327 = -3.814 of the x components of the vectors being added. The y component is similarly the sum -5.977 + 2.504 = -3.473 of the y components of the vectors. The magnitude of the resultant vector, by the Pythagorean Theorem, is therefore `sqrt( ( -3.814) ^ 2 + ( -3.473) ^ 2) = 5.158. The angle of the resultant vector to the x axis is arcTan( -3.473/ -3.814) = 57.29 degrees.
The final step depends on whether the x component -3.814 is greater or less than zero:
If -3.814 >= 0 then the following statement holds: Since the x component of this vector is positive, this is the correct angle.
If -3.814 <0 then the following statement holds: Since the x component of this vector is negative, the standard angle isµ 57.29 + 180 degrees="237.2" degrees.
Generalized Response: If we have vectors A and B, at angles `theta1 and `theta2 as measured from the direction of the positive x axis, their sum is found by multiplying the magnitude of each vector by the cosine of its angle to obtain its x component, and multiplying the magnitude of each vector by the sine of its angle to obtain its y component. The sum of the two x components will then be the x component of the resultant, and the sum of the two y components will be the y component of the resultant.
We obtain
Ax = |A| cos(`theta1),
Ay = |A| sin(`theta1),
Bx = |B| cos(`theta2),
By = |B| sin(`theta2).
We obtain the resultant vector R = A + B by first adding the x components of A and B:
Rx = Ax + Bx = |A| cos(`theta1) + |B| cos(`theta2)
and
Ry = Ay + By = A sin(`theta1) + B sin(`theta2).
We then find the magnitude and angle of R, using the Pythagorean Theorem and the arctangent:
|R| = `sqrt(Rx^2 + Ry^2)
and
`theta = arctan(Ry / Rx).
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Figure description: The figure below shows two vectors A and B, and their components Ax, Ay, Bx and By. The y components Ay and By are seen to add up to Ry, as the x components Ax and Bx are seen to at up to Rx. The magnitude and angle of the vector R are obtained by using the Pythagorean Theorem and the arctangent in the usual manner.
