Crossed up by Scalars and VectorsDate: 02/14/2012 at 06:21:31 From: Karthik Subject: Confusion about vectors. I have never really understood why dot products and cross products have been defined the way they are. Why is the result of a dot product a scalar, and that of a vector product a vector? I have consulted many books, and I have searched the net, but nobody seems to give any plausible explanation for this. Date: 02/14/2012 at 08:49:17 From: Doctor Jerry Subject: Re: Confusion about vectors. Hello Karthik, Thanks for writing to Dr. Math. Given the idea of a vector, it is natural to look for a way to calculate the angle t between vectors {a1, a2, a3} and {b1, b2, b3}. Letting ||a|| be the length of the vector a, if you apply the Law of Cosines to the triangle with sides a, b, and a - b, you will see (a1 - b1)^2 + (a2 - b2)^2 + (a3 - b3)^2 = a1^2 + a2^2 + a3^2 + b1^2 + b2^2 + b3^2 - 2||a||*||b||* cos[t] Simplify and you will find this, where "x y" means "x times y": a1 b1 + a2 b2 + a3 b3 = ||a||*||b||*cos[t] This makes it clear that the left side is a useful combination of a and b. For the cross product, if you have vectors a = {a1, a2, a3} and b = {b1, b2, b3}, it seems clear that a vector that is perpendicular to both of a and b would be useful. So, we seek a vector {c1, c2, c3} such that a.c = 0 b.c = 0 If you solve this system of two equations for c1 and c2, you will find c1 = -(-a3 b2 c3 + a2 b3 c3)/(a2 b1 - a1 b2) c2 = -(a3 b1 c3 - a1 b3 c3)/(a2 b1 - a1 b2) If you choose c3 as a2 b1 - a1 b2, this will give a solution and clean up the mess a bit. Please feel free to write back -- using the URLs at the bottom of this message -- if you have questions relative to my comments. - Doctor Jerry, The Math Forum http://mathforum.org/dr.math/ Date: 02/16/2012 at 03:07:00 From: Karthik Subject: Confusion about vectors. But could you please explain to me why the result of a dot product is a scalar and that of a cross product, a vector? Date: 02/16/2012 at 07:20:48 From: Doctor Jerry Subject: Re: Confusion about vectors. Hello, Thanks for writing to Dr. Math. 1. I said: "Given the idea of a vector, it is natural to look for a way to calculate the angle t between vectors {a1, a2, a3} and {b1, b2, b3}." The angle t or its cosine are scalars. 2. I also said: "For the cross product, if you have vectors a = {a1, a2, a3} and b = {b1, b2, b3}, it seems clear that a vector that is perpendicular to both...." The search starts with the idea of scalar and vector in these two situations. Please feel free to write back -- using the URLs at the bottom of this message -- if you have questions relative to my comments. - Doctor Jerry, The Math Forum http://mathforum.org/dr.math/ Date: 02/16/2012 at 11:35:29 From: Karthik Subject: Thank you (Confusion about vectors.) Thank you very much for all the help! |
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