X 2 x 2 contains only finitely many members of the set

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Unformatted text preview: c Spaces Some Exercises on Euclidean Spaces 1. Two points x and y of the space R k are said to be orthogonal to one another when x  y  0. Prove that if x and y are orthogonal to one another then x  y 2  x 2  y 2. Is the converse of this statement true? Since 106 xy the equation x  y 2 2 x 2  xy  xy  x  x  y  x  x  y  y  y  x  x  2x  y  y  y,  y 2 holds if and only if x  y  0. 2. Given any points x and y in R k , prove that x  y 2  x y 2  2 x 2  2 y 2. This identity is known as the parallelogram law. Is there also a parallelogram law for the Ý-norm? The parallelogram law follows at once when we expand the left side. The parallelogram law does not hold for the Ý-norm. For example 1, 0 0, 1 2  1, 0  0, 1 2 2 1, 0 2  2 0, 1 2 . Ý Ý Ý Ý 3. a. Prove that if x and y are points in R k and x  y  xy  1 then x  y. Hint: Look at the expression x y 2 . The hint says it all: x y 2 x y  x y  xx yx xyyy  1 1 1  1  0. It may be worth asking students to interpret this fact in terms of the angle between x and y as it appears in Exercise 4. b. Prove that if x and y are points in R k O and xy  x then there exists a positive number t such that x  ty. We define x t y and we observe that ty x   x x and we deduce from part a that x x which gives us x  ty. 4. If x and y are points in R k y x x  ty x 1 ty x O then we define the angle between x and y to be xy arccos . xy a. Why is the Cauchy-Schwarz inequality needed to make this definition make sense? The Cauchy-Schwarz inequality guarantees that xy 1. xy b. Prove that if  is the angle between two points x and y then x  y  x y cos . This equation follows at once from the definition of . c. What is the angle between two points x and y if these points are orthogonal to one another? When x  y  0 then the angle is arccos 0   . 2 107 5. Prove that if x and y are any points in R k then the points x  y and x y are orthogonal to one another if and only if x  y . Can you interpret this statement geometrically? The result follows at once from the equation xy  x y  x 2 y 2. A common interpretation of this exercise is that the diagonals of a parallelogram will be perpendicular to each other if and only if the parallelogram is a rhombus. Here is another interpretation: x y O x The angle subtended at any point on a circle by a diameter of the circle is a right angle. 6. In this exercise we suppose that a, b and c are points of R k and that a  b  c. Suppose that x  a  b  c. Prove that the points x a and b c are orthogonal to one another. Deduce two more similar statements and make a geometric interpretation of these statements that concerns the three altitudes of a triangle with vertices a, b and c. We see at once that x a  b c  a  b  c a  b c  b  c  b c  0. The point x is shown to be the common point of intrersection of the three altitudes of the triangle. 7. The cross product a b of two points a  a 1 , a 2 , a 3 and b  b 1 , b 2 , b 3 in...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.

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