II
Vectors Lecture and Vector Algebra
A set S of rays is called a direction if it satises 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 .
A vector A consists of a non-negative real number, called the magnitude of
the vector, and a direction. We denote the magnitude of A by |A|.
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
Let A by a vector, and let c by a real number. Multpying A by the scalar
c, we obtain a vector denoted by cA. The magnitude of the result is given
by |cA|
= |c||A|. The direction of cA is the same as the direction of A if
c 0, and opposite to the direction of A
if c < 0.
2 Addition of vectors
The sum of two vectors A and B is denoted simply by A + B . We can
dene this sum geometrically. Translate B such that its start point is the
end-point of A. Then A + B will be the vector having the same start-point
as A and the same end-point as B .
3 Scalar product (Dot product)
The scalar product of two vectors A and B is a scalar quantity denoted
by . A B . Its value is A . B = |A||B |cos, where is the angle made by A
and B if we translate B such that it has the same start-point as A. We
observe that |B |cos is the length of the projection of B on A.
1
B
A
A+B
Figure 1: Vector addition
The magnitude of A is equal to the square root of the dot product of A
with itself:
| A| =
Hence
A
|A|
= A
A .A
A .B
is a unit vector with the same direction as A.
4 Vector product (Cross product)
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:
|A B | = |A| |B |sin.
Let A and B have the same start-point P and end-points Q1 and Q2 ,
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 right-hand rule: if a right
hand is placed at P and the ngers are curling from P Q1 to T Q2 through
the angle smaller than , then the thumb indicates the direction of A B .
The following are some properties of these basic vector operations.
1. (a + b)C = aC + bC
2. (ab)C = a(bC )
3. a(B + C ) = aB + aC
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4. A + B = B + A
5. A . (B + C ) = A . B + A . C
6. A B = B A
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