High School: Number and Quantity
Vector and Matrix Quantities HSN-VM.B.4b
b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
If the directions of two added vectors are the same, they're easy to add. Just sum up the magnitudes. That's it.
Students should also know that to find the magnitude of two added vectors that don't have the same direction, we most often use the Law of Cosines.
For example, let's say that ||u|| = 10 and that ||v|| = 12, and that the angle between them is 60°. To find the resultant vector's magnitude, we use the Law of Cosines: c2 = a2 + b2 – 2ab cos(C). We could plug ||u|| = 10 for a, ||v|| = 12 for b, and 120° for ∠C. (In the problem, ∠C is the supplement of the angle connecting the vectors. That's where the 120° came from.)
If we do that, we'll have
c2 = 102 + 122 – 2(10)(12)cos(120°)
c2 = 100 + 144 – 240cos(120°)
c2 = 244 + 120 = 364
That means the addition of vectors u and v results in vector u + v where ||u + v|| ≈ 19.