 Hey there, today we're going to try to answer the question, what is a vector? In math and science, we see a lot of numbers, but a number by itself is meaningless. It only becomes a value when it is associated with a unit. The unit establishes a scale whereby the value can be compared to other values. The two together comprise a scalar value. One way to visualize a scalar value is to represent it in physical space on a number line. An effective number line will include two labeled tick marks. One that establishes the location of the starting number, usually zero, and often called the origin. The other mark establishes the magnitude, a scalar value with a number and a unit, represented by the distance between the two marks, as well as establishing in which direction the line is positive and which direction the line is negative or smaller values. For example, the distance between two tick marks on this number line represents, not 20 years, represents 10 years. The distance between two tick marks would represent 20 years. Once you have a number line, any scalar value using the same units can then be represented with the location on the number line. For example, my son is aged six, my age is 47, however, there's not enough particular room here, and my father's age, 40, 60. My father's age would be out here at about 68. Different units may represent the same measurable quantities. For example, length might be measured in centimeters or feet or miles. Time might be measured in seconds or days or decades. Weight might be measured in pounds or newtons or tons. If we use different units to measure the same quantities, if we use different units to measure the same quantities, there may be different values to represent the same scalar value. For example, one inch might be the same as 2.54 centimeters. Notice similar quantities, they're both lengths, but we're using different units to represent the same scalar value. Let's consider another term here. An entity is some identical object or construct with a dimension that can be measured with a quantity. In other words, a quantity measures a dimension of an entity. Let's consider some examples. Length is a quantity that measures across, that would be a dimension, a thumbnail. That would be the entity that would be measuring. Weight might be a quantity in earth gravity of a truck. The truck is the entity. And finally, time from start to finish of a swim race. The details about the dimension of the start to finish and the entity is the swim race. Note that multiple quantities measuring different dimensions can be associated with the same entity. For example, if the entity is Dr. Love, he might have multiple quantities associated with different dimensions. For example, his age, his weight, his height, and his heart rate are all different quantities that can be assigned scalar values. And notice the scalar values require units to give them meaning. BPM in this case is beats per minute, which is the count of the number of beats the heart would do in a period of a minute. This group, if we take this group of associated scalar values, that is known as a vector. The number of included scalar values is the dimension of the vector. In this case, this vector has four pieces of score scalar values associated with it. So it has a dimension of four. Notice another vector might exist with the same dimensions, but with different values, usually representing a different entity. For example, here's another person with a different age and a different weight, height, and heart rate. This ordered list of quantities defines the basis of the vector. In other words, this set of quantities, the age, the weight, the height, the heart rate are the basis of our vector. And notice the order that they are in is important. In algebra class, you learn about ordered pairs where it's important to list the first number followed by the second number. Similarly, if you're defining the basis for your vector, what the scalar values you're actually using or the quantities you're actually using are important, and the order that they're represented in is also important. So this is the basis while the associated numbers are the components. In this case, the components here are 6, 48, 49, and 78, whereas the components of the vector for the other entity for Dr. Love are 47, 203, 69, and 64. Note that the same vector can be expressed using different combination of basis and components. For example, in our original case up here, we had the information 47, 203, 69, and 64, which made sense in the basis where we had the age and years, the weight in pounds, the height in inches, and the beats per minute, which is the heart rate. However, if we took that information, we could have the same information reorganized. Consider this vector. In this case, the weight in pounds is listed first, the height is listed second, but it's now listed in centimeters. The heart rate is now listed in beats per hour, which is a strange number, but we'll use it in this case, and the age is listed in months. Notice that the components are different and the basis is different, but the vector is the same. It represents the same information. A number line, property labeled, is a visual representation of one dimension of the basis. Here, for example, is a representation of 47 years. Each component can be mapped with a location on the line. Here I'm establishing a basis of weight in pounds with a location representing a particular scalar value. This is yet another way of representing that vector of physical information for Dr. Love. If we take two of those number lines, something you've done in algebra classes, take two of those number lines and orient them perpendicularly. This creates a two-dimensional graphic basis with two components identifying a place in space. Notice this is a place in two-dimensional space. For example, this point in space here with the components 47 and 69 would represent someone who was 47 years old at the height of 69 inches. We can do a similar thing in three dimensions by using three different bases sketched out, in this case sketched out in two-dimensional space, but if we use those three dimensions, we can represent three pieces of information with the components 47, 203, and 64 here. We're representing somebody who is, again, 47 years of age, 203 pounds in weight, and has a heart rate of 64 beats per minute, as long as we're drawing this in the three-dimensional basis defined by these number lines.