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Math1300:MainPage/EuclideanSpace Contents • 1 Euclidean n-space ♦ 1.1 Defining Addition and Scalar Multiplication in ◊ 1.1.1 Theorem (Properties of Addition and Scalar Multiplication) • 2 The Dot Product in Euclidean n-space ♦ 2.1 Theorem (Properties of the Dot Product) ♦ 2.2 Theorem (Cauchy-Schwarz) ♦ 2.3 Corollary (Cauchy-Schwarz) • 3 Length in Euclidean n-space ♦ 3.1 Theorem (Triangle Inequality) ♦ 3.2 Theorem (Pythagorean Theorem) ♦ 3.3 Theorem (Length Properties) • 4 Distance in Euclidean n-space ♦ 4.1 Theorem (Properties of Distance in Euclidean n-space) Euclidean n-space We have derived many properties of 2-space and 3-space. In those cases we have studied ordered pairs (x0,y0) and ordered triples (x0,y0,z0). We now wish to generalize this concept to n-tuples. By this we mean that we will have n coordinates: a typical n-tuple is the set of all possible n-tuples by where v is a real number for We denote i Defining Addition and Scalar Multiplication in We define addition and scalar multiplication in If in a manner analogous to 2-space and 3-space: and r is any real number, then • • and Theorem (Properties of Addition and Scalar Multiplication) Let and A1: be vectors in n-space, and let r and s be real numbers (scalars). Then M1: is a vector A2: M2: A3: There exists a vector A4: is a vector For every vector A5: Contents M3: such that there exists a vector such that M4: M5: 1 Math1300:MainPage/EuclideanSpace Proof: A vector in may be viewed as a matrix. The rules for addition and scalar multiplication of vectors in and those for matrices are identical. Hence the proofs given for matrices carry through to vectors unchanged. The Dot Product in Euclidean n-space The dot product has been defined for 2-space and 3-space, and it is a straightforward concept to extend to if and then The length of a vector in n-space is defined analogously to that in 2-space or 3-space: The proofs for the different parts of the following theorem are virtually unchanged from those for 2-space and 3-space. Theorem (Properties of the Dot Product) • • • • if and only if • With the concepts of length and dot product defined, we may prove the following important theorem: Theorem (Cauchy-Schwarz) If and are vectors in then Proof: We want to consider the vector varies over all real numbers. Then The vectors and are considered fixed while x We define the function f(x) by and observe that f(x) is a polynomial of degree 2 and, since the function has at most one real root 2 (namely, 0). Remember that for any quadratic polynomial ax + bx + c, the roots are Theorem (Properties of Addition and Scalar Multiplication) In particular, 2 Math1300:MainPage/EuclideanSpace if b2 − 4ac > 0, then the quadratic polynomial has two real roots. This is exactly what doesn't happen here, and so which implies and Corollary (Cauchy-Schwarz) If and are vectors in then This implies that there is exactly one angle θ with so that that is, Notice that we now have the concept of both length and angle for even though we don't have the familiar geometry of 2-space and 3-space. In particular, we have a criterion to test whether or not two vectors are orthogonal: if and only if Length in Euclidean n-space Theorem (Triangle Inequality) If and are vectors in then Proof: The following figure indicates why this is called the triangle inequality. The length of one side of the triangle is less than or equal to the sum of the lengths of the other two sides. Theorem (Cauchy-Schwarz) 3 Math1300:MainPage/EuclideanSpace Theorem (Pythagorean Theorem) If and are vectors in then if and only if The following figure shows why this is called the Pythagorean theorem. Theorem (Length Properties) If is a vector in • • and r is any real number, then with equality if and only if Distance in Euclidean n-space With n > 3 we lose our usual geometric view of Euclidean n-space. All is not lost, however. We can often extend the ideas present in the cases where to higher values of n. The concept of distance in Euclidean n-space is one of those ideas. If and are two points, then we can think of the vector from P to Q as a vector using arrow notation. The same vector in coordinate notation is and the length of satisfies the distance from P to Q and denote it by d(P,Q). This means expressed in coordinate notation. Theorem (Pythagorean Theorem) We call this same value where P and Q are 4 Math1300:MainPage/EuclideanSpace Theorem (Properties of Distance in Euclidean n-space) Let , • • • and be vectors. Then with equality if and only if (Triangle inequality) Theorem (Properties of Distance in Euclidean n-space) 5