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.)
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}, v _{2}) 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 = (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:
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 .
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) | |
= | a i + b j + c k |
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.