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Download Contents The Arithmetic of Vectors The Length or Norm of a Vector
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Math1300:MainPage/MatrixNorm Contents • 1 The Arithmetic of Vectors ♦ 1.1 Theorem (Properties of Addition and Scalar Multiplication) • 2 The Length or Norm of a Vector in 2-space and 3-space ♦ 2.1 Theorem (Norm and Scalar Multiplication) The Arithmetic of Vectors Theorem (Properties of Addition and Scalar Multiplication) Let and A1: be vectors in 2-space or 3-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 M3: such that there exists a vector such that M4: A5: M5: Proof: A vector in 2-space or 3-space may be viewed as a matrix or a matrix. The rules for addition and scalar multiplication of vectors in coordinate notation and those for matrices are identical. Hence the proofs given for matrices carry through to vectors unchanged. The Length or Norm of a Vector in 2-space and 3-space The norm of a vector is its length. The length of a vector a vector in either arrow notation or coordinate notation. First we look at 2-space. Suppose that is denoted We want to compute the length of where A = (a ,a ) and B = (b ,b ). 1 2 1 2 The we use the Pythagorean theorem to compute the length: • Arrow notation: Contents Since length is nonnegative, we may 1 Math1300:MainPage/MatrixNorm write this as • Coordinate notation: Since length is nonnegative, we may write this as Next we consider 3-space. We simply use the Pythagorean theorem twice: In the x-y plane, the length of the vector (x,y,0) is and the length of satisfies Hence Theorem (Norm and Scalar Multiplication) For any vector and scalar r, Proof: In 2-space, let The Length or Norm of a Vector in 2-space and 3-space 2 Math1300:MainPage/MatrixNorm and so The proof for 3-space is essentially the same. Theorem (Norm and Scalar Multiplication) 3