Finding the Angle between Two Vectors
Date: 01/15/2006 at 19:39:55 From: Victor Subject: vector math There's two vectors A and B, which both have equal magnitudes. In order for the magnitude of A+B to be 120 times larger than the magnitude of A-B, what must the angle between them be?
Date: 01/16/2006 at 04:55:56 From: Doctor Luis Subject: Re: vector math Hi Victor, Using vector norm notation, the problem informs us that A and B are two vectors such that |A| = |B| Further, they want us to determine the angle T (between A and B) such that |A+B| = 120 * |A-B| Ok. Now that we have expressed the requirements in concise mathematical notation, let's solve the problem. The easiest way to find T is probably to use the dot product between A and B (denoted A.B). I'm sure you'll recognize the identity A.B = |A| * |B| * cos(T) Solving for cos(T) we get cos(T) = A.B/(|A| * |B|) if we use |A|=|B|, then cos(T) = (A.B)/|A|^2 Now, it is clear that the problem will be easier if we find the value (A.B) in terms of |A|^2. We can do that from the following relationship between the dot product and vector norm: v.v = |v|^2 (which is actually an instance of the identity above, applied to the same vector v, so that T=0, or cos(T)=1). Well the important thing is to realize that we can apply v.v = |v|^2 to |A+B|^2 and to |A-B|^2 (and then applying the distributive rule of the dot product), |A + B|^2 = (A + B).(A + B) = A.(A+B) + B.(A+B) = (A.A + A.B) + (B.A + B.B) = |A|^2 + 2(A.B) + |B|^2 = 2|A|^2 + 2(A.B) (using |A|=|B|) Similarly, |A - B|^2 = (A - B).(A - B) = A.(A-B) - B.(A-B) = (A.A - A.B) - (B.A - B.B) = |A|^2 - 2(A.B) + |B|^2 = 2|A|^2 - 2(A.B) (using |A|=|B|) Now, we'll use that second equation that the problem gave us: |A+B| = 120 |A-B| or |A+B|^2 = 120^2 * |A-B|^2 (2|A|^2 - 2(A.B)) = 120^2 * (2|A|^2 - 2(A.B)) You can use this last equation to solve for A.B in terms of |A|^2, which will allow you to find the ratio A.B/|A|^2 = cos(T), from which you can finally determine the value of T. As a sanity check, you should notice that the answer is small (at least relative to 180 degrees), which means that A and B are pointing in almost the same direction. This makes sense, since they'll reinforce each other when added, but almost cancel out when subtracted. This is how |A+B| can manage to be 120 times larger than |A-B|, even though the two vectors A and B have the same magnitude. I hope this helped! Let us know if you have any more questions. - Doctor Luis, The Math Forum http://mathforum.org/dr.math/
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