Problem: What are x and y the components of the vector obtained when we add vector A, with x and y components -6 and 3, to the vector B whose x and y components are 17 and -8? What are the magnitude and angle of this resultant vector?
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Solution: The x component is simply the sum -6 + 17 = 11 of the x components of the vectors being added. The y component is similarly the sum 3 + -8 = -5 of the y components of the added vectors. The magnitude of the resultant vector, by the Pythagorean Theorem, is therefore `sqrt( ( 11) ^ 2 + ( -5) ^ 2) = 12.08. The angle of the resultant vector to the x axis is arcTan( -5/ 11) = -24.43 degrees. Since the x component of this vector is positive, this is the correct angle.
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.
