In this section, we introduce a concept of key importance in many areas of math and science, namely vectors. So what is a vector? A vector is a quantity with two attributes, magnitude and direction, unlike ordinary numerical quantities, otherwise known as scalars or simply as numbers, which have only magnitude. Geometrically, a vector is simply an arrow, i.e. a line segment with a head indicating in which direction the arrow is pointing. Vectors can start and end anywhere.
Vectors live in a space with a certain number of dimensions, such as the two-dimensional xy-plane, otherwise known as R2 (read as "R squared"), where the symbol R is used to denote the real numbers. In this module, we will only concern ourselves with vectors in R2.
Vectors in R2 have two components. These are real numbers indicating the extent of a vector in the x and y-directions. Consider the vector v shown in the following figure. Since x changes by 3 in going from the tail to the head of v, the x-component of v is 3. Similarly, since y changes by 4 in going from the tail to the head of v, the y-component of v is 4. Since v is fully-described by its components, we may write it in the form v = (3,4). Note that this is the same notation we use for points. However, this should not cause any confusion because it is usually clear from the context whether we are talking about points or vectors.
Figure 6.7.1: Vector Components
A vector may be moved around in the xy-plane without being changed as long as it is kept pointing in the same direction and its length remains unchanged. Such motion as known as parallel transport. Thus, in the following figure, the two vectors shown are equal.
Figure 6.7.2: Parallel Transport
There are two elementary operations which may be performed on vectors, namely vector addition and scalar multiplication. Geometrically, the sum of two vectors u and v is a third vector w formed by first moving v so that its tail joins up with the head of u. The sum w = u + v is then the vector with its tail coinciding with the tail of u and its head with the head of v. This is shown in the following figure:
Figure 6.7.3: Vector Addition
It is easy to see that when vectors are added according to this definition, the components add, i.e. we have
- (6.7.4) (ux, uy) + (vx, vy) = (ux + vx, uy + vy)
Thus, in the example shown above we have (2,1) + (-3,4) = (-1,5).
Scalar multiplication just means multiplying a vector by an ordinary number. For instance, if v = (1,2) then 3v = (3.6). To generalize, if k is a scalar and v = (vx, vy) is a vector, then the scalar product kv is the vector given by
- (6.7.5) k(vx, vy) = (kvx, kvy)
In the above equation, k may be an arbitrary real number. If k = -1, then we write the result as -v = (-vx, -vy), known as the negative of the vector v. If k = 0, the result is the zero vector, 0 = (0,0).
The magnitude (also known as norm or length) of a vector is the distance from its tail to its head. The magnitude of a vector v is denoted as |v|. It is easy to see that the formula for the magnitude of the vector v = (vx, vy) is given by
A unit vector is a vector of magnitude 1. There are two important unit vectors, known as basis vectors, denoted by i and j. The components of these vectors are given by i = (1,0) and j = (0,1). The following diagram shows the basis vectors i and j.
Figure 6.7.7: The Basis Vectors i and j
Every vector has a unique representation as a sum of scalar multiples of i and j. Specifically we have
- (6.7.8) v = (vx, vy) = vxi + vxj
This is an alternate way of writing a vector in terms of its components.
Normalization of a vector means dividing it by its magnitude. Thus, the normalization of a vector v is given by v / |v|. The resulting vector is the unit vector pointing in the same direction as v.
Example 1: Determine the magnitude and normalization of the following vectors:
- (a) (2, 0)
- (b) (0,- 5)
- (c) (3, 4)
- (d) (1, 1)
- (e) (-5, 7)
- (a) The magnitude of the vector (2, 0) is given by √22 + 02 = 2. Its normalization is (2, 0) / 2 = (1, 0).
- (b) The magnitude of the vector (0, -5) is given by √02 + (-5)2 = 5. Its normalization is (0, -5) / 5 = (0, -1).
- (c) The magnitude of the vector (3, 4) is given by √32 + 42 = 5. Its normalization is (3, 4) / 5 = (3/5, 4/5) = (0.6, 0.8).
- (d) The magnitude of the vector (1, 1) is given by √12 + 12 = √2. Its normalization is (1, 1) / √2 = (√2/2, √2/2).
- (e) The magnitude of the vector (-5, 7) is given by √(-5)2 + 72 = √74. Its normalization is (-5, 7) / √74 = (-5/√74, 7/√74).