Law of cosines method for adding two vectors

If you know the magnitudes of two vectors and the angle between them, this is an efficient method.

The sketch on the left shows vectors of magnitudes A and B. On the right, R is the vector sum at angle a from B. Note that q + f = 180^o, so cosf = - cosq . Write the law of cosines involving cosf (see below for more on this law), replace it with -cosq , and show that

R = (A² + B² + 2ABcosq )^1/2

Also note that a = tan^-1[Asinq /(B+Acosq )]

More on the law of cosines:

It is the Pythagorean Theorem with an adjustment: The square of any side of a triangle is equal to the sum of the squares of the other two sides minus an adjustment. The adjustment is twice the product of the other two sides times the cosine of the angle between them. This reduces to the Pythag Thm when the angle between them is 90^o, because cos 90^o = 0.

I just noticed a trig identity that falls out of this. When vectors A and B are equal (see sketch above) a is q /2, so from our equation for a we get

tan(q /2) = sinq /(1+cosq ).

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Comments, questions: fredgram