Problem: What are x and y the components of the vector obtained when we add vector A, with x and y components 6 and -13, to the vector B whose x and y components are -12 and 2? What are the magnitude and angle of this resultant vector?
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Solution: The x component is simply the sum 6 + -12 = -6 of the x components of the vectors being added. The y component is similarly the sum -13 + 2 = -11 of the y components of the added vectors. The magnitude of the resultant vector, by the Pythagorean Theorem, is therefore `sqrt( ( -6) ^ 2 + ( -11) ^ 2) = 12.52. The angle of the resultant vector to the x axis is arcTan( -11/ -6) = 61.36 degrees. Since the x component of this vector is negative, the standard angle is 61.36 + 180 degrees = 241.3 degrees.
Generalized Response: Vectors A and B, with their x and y components indicated, are shown on the figure below. The sum of the two x components will be the x component of the resultant, and the sum of the two y components will be the y component of the resultant:
Rx = Ax + Bx
and
Ry = Ay + By.
We 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.
