1873_solutions

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 xy the equation x  y 2 2 x 2  xy  xy  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  xy  1 then x  y. Hint: Look at the expression x y 2 . The hint says it all: x y 2 x y  x y  xx yx xyyy  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 xy  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 xy arccos . xy a. Why is the Cauchy-Schwarz inequality needed to make this definition make sense? The Cauchy-Schwarz inequality guarantees that xy 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 xy  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...
View Full Document

## 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.

Ask a homework question - tutors are online