Introduction To Vectors

Vectors are commonplace in many areas of study, specifically physics. In math courses up to this point, you have likely worked with numbers that represent a magnitude of some sort. Things such as temperature, profits, and expenses are all examples of such quantities. From this point on, we will refer to these types of values as scalar values. They are values that have just a magnitude associated with them.

Vectors, on the other hand, involve both a magnitude like scalars, as well as a direction that the magnitude is being applied in. Adding the direction allows us to be able to consider how things like forces interact with each other, when they are applied in the same direction, opposite of each other, or maybe some combination of those.

To work with vectors, we need to first understand a little bit about how they interact with each other. For now, let’s imagine vectors in the (x,y) coordinate plane. We will depict vectors as lines, with an arrow pointing in the direction they travel in, as shown below.

We will denote a vector using a symbol with an arrow above it, like this: \vec{x}. If we are referring to the magnitude of the vector, we surround it in absolute value bars, like this: |\vec{x}|.

Example: Suppose that we travel north at a rate of 55 km/h. Let \vec{x} be the vector that represents our travel velocity. What is the direction and magnitude of \vec{x}.

In this case, the direction of the vector is north, as stated in the example. The magnitude is |\vec{x}| = 55 km/h.

The system of vectors is rather intuitive from a physical perspective, it makes sense to include direction and magnitude in instances like velocity which are common to physics. At this point you may wonder how we mathematically represent vectors.

The cartesian plane is naturally good at representing vectors that are in two dimensions. Since direction and magnitude can be represented by tweaking the x and y coordinates, we can easily depict and manipulate vectors using this coordinate system. From here, we can start to understand how vectors interact with vectors, and how scalars interact with vectors.

Example: Suppose we have a vector \vec{a} = (2,3) and \vec{b} = (5,4). Determine the value of \vec{a} + \vec{b}

Adding vectors is as simple as adding their components. Therefore \vec{a}+\vec{b} = (7,7).

The reason why we can just add the x and y components to each other relates to the geometry of our cartesian system. Consider two vectors, \vec{a} and \vec{b}, which move in two arbitary directions.

From this diagram, you can see what adding two vectors looks like in a geometric sense. From this, we can see that the components being added together is an accurate depiction of vector addition.

In addition, keep in mind that subtraction is simply adding the negative. This means that we can apply the same addition rules to subtraction as well, meaning that subtraction is just subtracting each component.

Example: Suppose that we have a vector \vec{a} = (1,5). Determine the value of 2* \vec{a}.

This operation is called scalar multiplication. This operation is equivalent to adding two of the vector \vec{a} together in this example. From this, we can see that we can logically multiply both components by 2 to get the right answer. Therefore, 2 * \vec{a} = (2,10).

Before we move forward, I want to first discuss the cartesian system, and how it applies to vectors. Notice that we are working with the (x,y) coordinate system. These coordinates are always real numbers. If you have taken some math courses already, you will know that we denote real numbers with the symbol \mathbb{R}. When we talk about the coordinate system, (x,y), we are discussing the number system \mathbb{R}^2, which is simply a 2-dimensional number system.

I bring this up since vectors are not limited to existing in two dimensions. Often, we see them written in three dimensions, or \mathbb{R}^3. These vectors have components (x,y,z). We can continue to increase the dimensions to any \mathbb{R}^n, where n is any integer. We will see these numbering systems throughout linear algebra.

With this information, we can now start to discuss the properties of vectors.

Properties of Vector Addition and Scalar Multiplication: Suppose that \vec{a}, \vec{b}, \vec{c} are vectors, and s,t are some scalars in the real number set. The following properties hold:

  1. \vec{a}+\vec{b} = \vec{b}+\vec{a}
  2. (\vec{a}+\vec{b})+\vec{c} = \vec{a}+(\vec{b}+\vec{c})
  3. There exists a vector \vec{0}, such that \vec{a}+\vec{0} = \vec{a}
  4. For each vector \vec{a}, there exists a vector \vec{b}, such that \vec{a}+\vec{b} = \vec{0}
  5. 1*\vec{a} = \vec{a}
  6. (s*t)\vec{a} = s*(t*\vec{a})
  7. s*(\vec{a}+\vec{b}) = s*\vec{a}+s*\vec{b}
  8. (s+t)\vec{a} = s*\vec{a} + t*\vec{a}

Most of these properties are rather easy to intuitively see with a little understanding of how numbers, in general, add and multiply. I will show an example of proving property 1.

Proof of Property 1: \vec{a} + \vec{b} = \vec{b} + \vec{a}

If we write this in component form, you get:

(x_1,x_2,x_3,...x_n) + (y_1,y_2,y_3,...,y_n) = (y_1+y_2+y_3+...+y_n)+(x_1,x_2,x_3,...x_n)

Since the components are a part of the real numbers, we know that x_1+y_1 = y_1+x_1, therefore the property holds.

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