Problem: What are the magnitude and angle of a vector whose x and y components are respectively 3 and 16?
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Solution: The magnitude of the vector is found by the Pythagorean Theorem to be `sqrt( ( 3) ^ 2 + ( 16) ^ 2) = 16.27. The angle made by the vector with the x axis is arctan ( 16/ 3) = 79.35 degrees. Since the components of the vector are both positive, the vector is in the first quadrant. Its angle with the positive x axis is therefore equal to the 79.35 degrees found above.
Generalized Response: The components vx and vy correspond to the sides of a right triangle whose hypotenuse is equal to the length or magnitude of the vector, and whose angle with the positive x axis is opposite to the y component and adjacent to the x component.
The magnitude of the vector is therefore found from the Pythagorean Theorem to be v = `sqrt(vx^2 + vy^2).
The angle has a tangent equal to vy / vx, so the angle is the arctangent arcTan(vy / vx).
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Figure description: The figure below depicts a vector with components vx and vy. By the Pythagorean Theorem the magnitude of the vector is `sqrt(vx^2 + vy^2). By the definition of the tangent, the angle with the positive x axis is arcTan(vy / vx).
