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Home : Math & Science : Physics Study Guides : Vectors : Addition : The Component Method for Vector Addition and Scalar Multiplication
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.)
Figure 1.1: The vector (a, b) in the Euclidean plane.
Vector Addition Using Components
| u + v = (u1 + v1, u2 + v2) |
For three-dimensional vectors u = (u1, u2, u3) and v = (v1, v2, v3), the
formula is almost identical:
| u + v = (u1 + v1, u2 + v2, u3 + v3) |
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.
Scalar Multiplication of Vectors Using Components
Given a single vector v = (v1, 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:
| av = (av1, av2) |
Similarly, for a 3-dimensional vector v = (v1, v2, v3) and a scalar a, the
formula for scalar multiplication is:
| av = (av1, av2, av3) |
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, and k, 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:
| (a, b, c) | = | a(1, 0, 0) + b(0, 1, 0) + c(0, 0, 1) | |
| = | ai + bj + ck |
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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