By Serguei A. Stepanov

Writer S.A. Stepanov completely investigates the present country of the idea of Diophantine equations and its comparable tools. Discussions specialise in mathematics, algebraic-geometric, and logical features of the challenge. Designed for college students in addition to researchers, the publication contains over 250 excercises observed through tricks, directions, and references. Written in a transparent demeanour, this article doesn't require readers to have exact wisdom of recent tools of algebraic geometry.

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**Example text**

THE CROSS PRODUCT 37 This triangle is obtained by connecting the three points with lines. Picking (1, 2, 3) as a starting point, there are two displacement vectors, (−1, 0, 2) and (4, −1, −1) such that the given vector added to these displacement vectors gives the other two vectors. The area of the triangle is half the area of the parallelogram determined by (−1, √ −1) . Thus (−1, 0, 2) × (4, −1, −1) = (2, 7, 1) √ 0, 2) and (4, −1, and so the area of the triangle is 21 4 + 49 + 1 = 32 6. 7 In general, if you have three points (vectors) in R3 , P, Q, R the area of the triangle is given by 1 |(Q − P) × (R − P)| .

6). In the case where b is not a multiple of a, it follows f (t) > 0 for all t which says f (t) has no real zeros and so from the quadratic formula, 2 2 2 (2 (a · b)) − 4 |a| |b| < 0 which is equivalent to |(a · b)| < |a| |b|. 5). This means that whenever something satisﬁes these properties, the Cauchy Schwarz inequality holds. There are many other instances of these properties besides vectors in Rn . The Cauchy Schwarz inequality allows a proof of the triangle inequality for distances in Rn in much the same way as the triangle inequality for the absolute value.

For u, v vectors, consider (u − projv u) · (u − projv u) ≥ 0 Now simplify using the axioms of the dot product and then put in the formula for the projection. Of course this expression equals 0 and you get equality in the Cauchy Schwarz inequality if and only if u = projv u. What is the geometric meaning of u = projv u? 11. A boy drags a sled for 100 feet along the ground by pulling on a rope which is 20 degrees from the horizontal with a force of 40 pounds. How much work does this force do? 12.