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Unformatted text preview: Lecture II
Vectors and Vector Algebra A set S of rays is called a direction if it satisﬁes the following laws:
(1) Any two rays in S have the same direction.
(2) Every ray that has the same direction as some member of S is in S .
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A vector A consists of a nonnegative real number, called the magnitude of
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the vector, and a direction. We denote the magnitude of A by A.
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A vector A such that A = 1 is called a unit vector.
We will now describe four basic algebraic operations with vectors in E3 :
1 Multiplication by a scalar
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Let A by a vector, and let c by a real number. Multpying A by the scalar
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c, we obtain a vector denoted by cA. The magnitude of the result is given
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by cA = cA. The direction of cA is the same as the direction of A if
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c ≥ 0, and opposite to the direction of A if c < 0.
2 Addition of vectors
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The sum of two vectors A and B is denoted simply by A + B . We can
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deﬁne this sum geometrically. Translate B such that its start point is the
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endpoint of A. Then A + B will be the vector having the same startpoint
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as A and the same endpoint as B .
3 Scalar product (Dot product)
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The scalar product of two vectors A and B is a scalar quantity denoted
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by A . B . Its value is A . B = AB cosθ, where θ is the angle made by A
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and B if we translate B such that it has the same startpoint as A. We
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observe that B cosθ is the length of the projection of B on A. 1 �
B �
A
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A+B Figure 1: Vector addition
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The magnitude of A is equal to the square root of the dot product of A
with itself:
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 A =
Hence �
A
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A �
= √A ��
A .A �
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A .B is a unit vector with the same direction as A. 4 Vector product (Cross product)
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The cross product of two vectors A and B is a vector denoted by A × B .
The magnitude of the cross product is given by:
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A × B  = A B sinθ.
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Let A and B have the same startpoint P and endpoints Q1 and Q2 ,
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respectively. Let M be the plane of A and B . The direction of A × B is
normal to M in the manner established by the righthand rule: if a right
hand is placed at P and the ﬁngers are curling from P Q1 to T Q2 through
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the angle smaller than π , then the thumb indicates the direction of A × B .
The following are some properties of these basic vector operations.
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1. (a + b)C = aC + bC
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2. (ab)C = a(bC )
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3. a(B + C ) = aB + aC 2 ��
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4. A + B = B + A
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5. A . (B + C ) = A . B + A . C
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6. A × B = −B × A 3 ...
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This note was uploaded on 09/24/2011 for the course MATH 1802 taught by Professor Duorg during the Two '04 term at Macquarie.
 Two '04
 Duorg
 Math, Algebra, Vectors

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