Find the magnitude and direction of 101m,63.5m,57.0

  158.085∠69.6°

Explanation:

The most straightforward way to work these is to convert them to rectangular coordinates and back. It looks like you may have several problems of this nature, so it may work well for you to do the calculations using a spreadsheet. That way, you can just change the numbers for a new problem, and not have to mess with the formulas.

For magnitude m at angle a, the rectangular coordinates are ...

  m∠a = (x, y) = (m·cos(a), m·sin(a))

For these two vectors, the rectangular coordinates are ...

  A = (0, 63.5)

  B = (101·cos(57°), 101·sin(57°)) ≈ (55.0085, 84.7057)

Then the sum is ...

  A + B = (0 +55.0085, 63.5 +84.7057) = (55.0085, 148.2057)

___

The conversion back to polar coordinates is ...

  m = √(x²+y²)

  a = arctan(y/x) . . . . paying attention to quadrant based on the signs*

For this sum,

  m = √(55.0085² +148.2057²) ≈ √24990.9 ≈ 158.085

  a = arctan(148.2057/55.0085) ≈ arctan(2.69423) ≈ 69.6°

So, the sum is ...

  A + B ≈ 158.085∠69.6°

* If you do this using a spreadsheet, the ATAN2 function chooses the quadrant based on the signs of the two arguments. Note that results may be in radians, so may require conversion to degrees. (You may also have to convert degrees to radians for the sine and cosine functions.)

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