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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 Ýnorm?
The parallelogram law follows at once when we expand the left side. The parallelogram law does
not hold for the Ýnorm. 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 CauchySchwarz inequality needed to make this definition make sense?
The CauchySchwarz 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.
 Fall '08
 STAFF
 Math, Calculus

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