a b OB
or B b
OA AB OB C
B
parallelogram law
b
O
a
A
O
a
A
6
The properties of addition
The addition of vectors satisfies the following laws: (1) Commutative law a b b a; (2) Associative law (a b) c a (b c); (3) a 0 a; (4) a ( a) 0. Subtraction:
then it represents a point, and it is also the only vector with no specific direction.
In textbooks, vectors are usually written in lowercase, boldface letters, for example, u,v and w. Sometimes we use uppercase boldface letters, such as F, to denote a force vector. In handwritten form, it is customary u , v , w and F . to draw small arrows above the letters, for example
direction is the same as that of a is written as a . A vector whose length is 0 is called the zero vector and is written as 0. Since the initial point of the zero vector coincides with its terminal point,
a-b=a+( - b)
b
a b b a a
c
a+b+c
7
(2) Scalar Multiplication
Definition Scalar Multiplication Let l be a nonzero scalar and a a nonzero vector. Then the product (or scalar multiple) of l and a is a vector, denoted by la. Its length is ||la||=|l|||a||, its direction is the same as that of a if l>0 or is opposite to that of a if l<0. If l=0 or a=0, we define la=0. By this definition, we have ( 1)a a,
speed of the moving object.
2
The Concept of Vector
The directed line segment AB has initial point A and terminal point B ; its length is denoted by || AB || . Directed line segments that have the
the action in terms of a suitably chosen
unit. For example, a force vector points in the direction in which it is applied and
its length is a measure of its strength; a velocity vector points in the direction of motion and its length is the
Suppose that a1 ,a2 ,,ak ( k 3) are k vectors with a common initial point. If they lie in the same plane, then we say that these vectors
are coplanar. It is easy to see that any two vectors are coplanar.
Terminal point
same length and direction are equivalent. y
B
Initial point
AB
O
x
A
Definition
Vector, Equal Vector
A vector in the plane is a directed line segment. Two vectors are equal
Lecture 4
Vectors and Their
Operations
The Concept of Vector
A quantity such as force, displacement, or velocity is represented by a directed line segment. The arrow points in the direction of the action and its length gives the magnitude of
The scalar product satisfies the following laws: (1) (2) (3) Commutative law Associative law Distributive law
l(ma)=m (la) l(ma)=(lm)a l(a+b)=la+lb
(l+m)a=la+ma
AMB (not greater than p) is called the
b
B
included angle between the vectors a and b, denoted by (a,b). If the included angle p between a and b is , then a and b are 2 said to be perpendicular, denoted by a b.
5
Linear Operations on Vectors (1) Addition of Vectors
Definition Triangle law of addition of vectors Suppose a and b are any two vectors and O is any point. If we draw a vector OA a from O to A and then draw the vector AB b starting from point A of a, then the vector OB is called the sum of a and b, denoted by a b, that is,
(or the same) if they have the same length and direction.
3
The Concept of Vector
A vector whose length is 1 is called a unit vector; a unit vector whose
If two vectors a and b have same or opposite direction, we say that
they are parallel or collinear, denoted by a // b.
The vectors a and b are said to be orthogonal or perpendicular, if their directions are orthogonal, and denoted by a b.
and a || a || a , where a is the unit vector
with the direction of a, so that we have a a , a 0. || a ||
8
The properties of Scalar Multiplication
proja b || b || cosq a
is called the orthogonal projection vector of b onto a (or onto the unit vector with the same direction as a), or simply the projection vector. The scalar
M
q
A
a
11
Projection of Vectors
Definition Orthogonal projection vector, orthogonal projection Suppose that the included angle between the vectors a and b is q. Then the vector
(4)
1a=a.
9
The length of a Vectors
By the above discussion we known that the length of a vector has the following basic properties: (1) (2) (3) Nonnegativity || a || 0, and || a || 0 a 0; Absolute homogeneity || l a ||| l ||| a ||; Triangle inequality || a b |||| a || || b ||, where the sign of equality holds iff a and b have the same direction. The geometric meaning of the triangle inequality is that the sum of the lengths of two adjoining sides of a triangle is greater than or equal to the length of the third side of the triangle.