Skip to main content

Bring Me Vector - Basics

via GIPHY

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.


Comments

Popular posts from this blog

The Purpose

  The sun was just about to set. I reached just then. I got down from the car and walked to the river. I kneeled down and put my hand in the water. The water in the river had a little warmth because it was a sunny day. I sat down there, waving my hand for some time. It was so good, I hoped the time stopped then and there. Then I thought, my hand was feeling the experience of water. Irrespective of which part of the river I put my hand in, the experience will remain mostly stay the same. If I enter the river and swim, then my experience would become more vivid and significant. If I submerge myself completely underwater then the experience of water around me becomes everything. Hypothetically, just imagine because of my sheer desire to stay there I obtain the ability to breathe and live underwater. I would probably choose to explore further rather than swim back to the surface. I will start imitating any creature I would find living in that water. Slowly I will learn a movement of my own

Chaos - Discarding The Past

It gets hard. Hard to preserve the knowledge and wisdom of the past. Amidst indefinite human stupidity. And one generation will surely and completely stop passing it down. From that day on, Humanity will evolve in a certain direction rapidly. But will lose a connection to their past so important and fundamental to their existence, it becomes almost impossible to revive that because of the dimension they have evolved along. Today, hormones travel faster than thoughts. We were optimized to move in that direction. Because, there is means. Means to deliver the message. Message to condition the mind of people. Our minds reward us for bearing garbage, but will penalize, if we even try to contemplate on our own philosophical and moral standpoints. Who would find time to appreciate the un countable and thorough trials and errors out ancestors had to go through arrive at conclusions about the significance of various days in any year. Who would feel proud, that out ancestors had studied and put

Beyond Words

  The yearning, The desire. It's hard to tell if it is a selfless pursuit or a selfish endeavor. I never understood the great emotion being confined to a few words. Sometimes it's just the pleasantness of the union. Sometimes it's the pain of separation. Every other emotion feels like a by-product of that. It devolves into something evil the moment it's not about people but things.  In its presence, all things sweet don't feel the same always. And all sad things don't stay the same. When it is understood deeply, pleasure will never be in taking, but in giving. It is what gives comfort even inside a prison of our own making. It is the warmth in one's association. It is also the warmth of one's tears flowing down on their cheeks on farewell. Is there somewhere that it isn't there. I don't need to celebrate it particularly if I cherish it every day in one form or another. What it symbolizes neither changes for anyone nor anything. It is misinterpret