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6.7. Vectors

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

*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.*

**scalars**Vectors live in a space with a certain number of dimensions, such as the two-dimensional xy-plane, otherwise known as **R**^{2} (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 **R**^{2}.

Vectors in **R**^{2} 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

*. Geometrically, the sum of two vectors*

**scalar multiplication****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)**(u_{x}, u_{y}) + (v_{x}, v_{y}) = (u_{x}+ v_{x}, u_{y}+ v_{y})

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 3**v** = (3.6). To generalize, if k is a scalar and **v** = (v_{x}, v_{y}) is a vector, then the scalar product k**v** is the vector given by

**(6.7.5)**k(v_{x}, v_{y}) = (kv_{x}, kv_{y})

In the above equation, k may be an arbitrary real number. If k = -1, then we write the result as -**v** = (-v_{x}, -v_{y}), 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

*or*

**norm***) of a vector is the distance from its tail to its head. The magnitude of a vector*

**length****v**is denoted as |

**v**|. It is easy to see that the formula for the magnitude of the vector

**v**= (v

_{x}, v

_{y}) is given by

**(6.7.6)**

A * unit vector* is a vector of magnitude 1. There are two important unit vectors, known as

*, denoted by*

**basis vectors****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**= (v_{x}, v_{y}) = v+ v_{x}i_{x}**j**

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)

**Solution:**

- (a) The magnitude of the vector (2, 0) is given by √2
^{2}+ 0^{2}= 2. Its normalization is (2, 0) / 2 = (1, 0). - (b) The magnitude of the vector (0, -5) is given by √0
^{2}+ (-5)^{2}= 5. Its normalization is (0, -5) / 5 = (0, -1). - (c) The magnitude of the vector (3, 4) is given by √3
^{2}+ 4^{2}= 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 √1
^{2}+ 1^{2}= √2. Its normalization is (1, 1) / √2 = (√2/2, √2/2). - (e) The magnitude of the vector (-5, 7) is given by √(-5)
^{2}+ 7^{2}= √74. Its normalization is (-5, 7) / √74 = (-5/√74, 7/√74).

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