Bring Me Vector - Basics

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In this article, I`m going to explain the very basics of vectors. Then in the upcoming articles, I`m going to explain more concepts like span, subspace, basis vectors, vector components... etc.

"A vector is a geometrical object with both magnitude and direction. In simple words, it's a directed line segment, from two points A to B, where the line segment's length is the vector's magnitude and the line's direction from A to B is the direction of the vector AB. "

Well, that's quite a mouthful. I would advise the readers through my previous blog article What`s The Point? to understand more about the concepts of point and line.

Now, let us try to understand the above-provided definition of a vector by understanding line segments. A line segment is a line between two points. And a line is nothing but an infinite set of points. So ultimately a line segment is an infinite set of points between two points. 

Now imagine a two-dimensional space where we have identical line segments scattered all over the space.

Identical How? you may ask me. Because all of those line segments rise 1-unit with respect to the y-axis and run 1-unit with respect to the x-axis. Every line segment has a starting point and an ending point. Let us call the starting point as tail and the ending point as head. Now if you observe, in the above two-dimensional space, every line segment`s tail and head are different. i.e. the cartesian points of the head and tail are different for all the line segments. But the idea that we are more interested in here is the direction of the line segment. All the line segments in the above two-dimensional space are pointing in the same direction. In that sense that they are pointing in the same direction and have the same rise and run with respect to the two-dimensional space we can call these line segments identical.

To understand the idea of head and tail, you can go through the below diagram.

But an important thing to know is that the head B(1,1) is not equivalent to the cartesian point (1,1). Because the line segments tail is not starting from the origin i.e (0,0).

When a line segment`s tail is the origin (0,0), then that line segment or vector is said to be in the standard position. Also, AB is said to be the positional vector of point B when point A is the origin.

So a vector that is in standard position can be simply represented by its head, as below.

A vector can be represented in two ways, a row vector, or a column vector.

The magnitude of the vector is nothing but the length of the line segment. And it can be calculated by using the Pythagoras theorem.

And to find out the magnitude of a vector that is not in the standard position, we can simply use the formula used to calculate the distance between two points.

As discussed in the previous article What`s The Point? Adding a dimension means adding a degree of freedom to the point. Imagine the vector is present in the standard position, the dimensionality of a vector is the number of elements in the vector.

In a one-dimensional space (number line), the dimensionality of the vector below is 1, And in a two-dimensional space, the dimensionality of the vector below is 2.

And similarly, in a three-dimensional space, the dimensionality of the below vector is 3.