Download Contents The Arithmetic of Vectors The Length or Norm of a Vector

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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
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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
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and so
The proof for 3-space is essentially the same.
Theorem (Norm and Scalar Multiplication)
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