The
Component Method for Vector Addition and Scalar Multiplication
When we mentioned in the introduction that a vector is
either an ordered pair or a triplet of numbers we implicitly defined vectors in
terms of components.
Each entry in the
2-dimensional ordered pair (a, b) or 3-dimensional
triplet (a, b, c) is called a
component of the vector. Unless otherwise specified, it is normally understood
that the entries correspond to the number of units the vector has in the x , y ,
and (for the 3D case) z directions of a plane or space. In other words, you can
think of the components as simply the coordinates of the point associated with
the vector. (In some sense, the vector is the point, although
when we draw vectors we normally draw an arrow from the origin to the point.)
Vector
Addition Using Components
Given two vectors u =
(u 1, u 2) and v =
(v 1, v 2) in the Euclidean
plane, the sum is given by:
|
u + v =
(u 1 + v 1, u 2 + v 2)
|
|
For
three-dimensional vectors u = (u 1, u 2, u 3) and v =
(v 1, v 2, v 3) ,
the formula is almost identical:
|
u + v =
(u 1 + v 1, u 2 + v 2, u 3 + v 3)
|
|
In
other words, vector addition is just like ordinary addition: component
by component.
Notice that if you add
together two 2-dimensional vectors you must get another 2-dimensional vector as
your answer. Addition of 3-dimensional vectors will yield 3-dimensional
answers. 2- and 3-dimensional vectors belong to different vector spaces and
cannot be added. These same rules apply when we are dealing with scalar
multiplication.
Given a single vector v =
(v 1, v2) in the Euclidean plane,
and a scalar a (which is a real number), the multiplication of
the vector by the scalar is defined as:
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av =
(av 1, av 2)
|
|
Similarly, for a
3-dimensional vector v = (v 1, v 2, v 3) and
a scalar a , the formula for scalar multiplication is:
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av =
(av 1, av 2, av 3)
|
|
So what we are doing when
we multiply a vector by a scalar a is obtaining a new vector
(of the same dimension) by multiplying each component of the
original vector by a .
Unit
Vectors
For 3-dimensional vectors,
it is often customary to define unit vectors pointing in the x , y ,
and z directions. These vectors are usually denoted by the
letters i , j , andk ,
respectively, and all have length 1 . Thus, i = (1,
0, 0) , j = (0, 1, 0) , and k = (0,
0, 1). This enables us to write a vector as a sum in the following way:
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(a, b, c)
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=
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a(1,
0, 0) + b(0, 1, 0) + c(0, 0, 1)
|
|
|
|
=
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a i + b j + c k
|
|
Vector
Subtraction
Subtraction for vectors (as
with ordinary numbers) is not a new operation. If you want to perform the
vector subtraction u - v , you simply use the
rules for vector addition and scalar multiplication: u - v = u +
(- 1)v .
In the next section,
we will see how these rules for addition and scalar multiplication of vectors
can be understood in a geometric way. We will find, for instance, that vector
addition can be done graphically (i.e. without even knowing the components of
the vectors involved), and that scalar multiplication of a vector amounts to a
change in the vector's magnitude, but does not alter its direction.
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