A Mathematical Blog

(Note : please patiently allow all 3D applets to load. Depending on your internet speed, the page might take a little longer to load. Also, with dark mode on, you might not be able to see math equations in some browsers.) I have already talked about direction angles in great detail in my previous post. If you are here to read about the essentials I’ll suggest you should read my previous post.  In this short post I want to show you a neat little visualisation, something which, despite my best efforts, I couldn't squeeze into earlier post. I’ll intend to keep it short and to the point as possible, because I have already written a great deal about direction angles. So let’s get straight to it. Firstly, let me state the definition(sort of). Direction angles of a vector are the three angles made by the vector with positive arms of X, Y and Z axes.  Direction Angles of   :   60°, 45 ° , 60 °           \(\vec{u}\)  Direction angles are a way to express the direction of a vecto

Direction of a Vector in 3D My previous post was about two primary ways in which we can describe the direction of a vector in two dimensional space. In this post, we will add in a third dimension(Z-axis) and discuss about a number of titles that are used to describe the direction of a vector in three dimensional space.  Let’s take an example of a vector in 3D in component form, \(\vec{u} = 2\hat{i} + \frac{4}{\sqrt{2}} \hat{j} + 2\hat{k}\) Just like for a 2D vector, the scalar components of a 3D vector can tell us the direction of the vector in the coordinate system. From the origin, we go 2 units in the positive X-direction; take a right angle turn and travel \(\frac{4}{\sqrt{2}}\) units(about 2.8 units) in the positive Y-direction; and finally travel 2 units in the positive Z-direction.   The arrow drawn from the start(origin) to the end points us in the direction of \(\vec{u}\). So a description that properly describes the direction(& magnitude) of \(\vec{u}\) could be, 

Describing Direction of a vector in 2D (If you want to read a post on how to describe the direction of a vector in 3D which also includes Direction angles and direction cosines, click here )  There are many ways to specify the direction of a vector in 2D, depending on what information we are given about it. Let \(\vec{v}\) be a vector given in component form,  \(\vec{v} = 2\hat{i} + 2\hat{j}\) We could simply say that \(\vec{v}\) is pointing somewhere in north-east direction to kinda roughly describe the direction. But why give a sketchy account of direction when there are simple and better ways which allow us to precisely describe the direction for a vector given in component form.  One way to perfectly describe the direction of a vector, is from its two scalar components, 2 and 2. If we choose origin as the tail of the vector arrow representing \(\vec{v}\), then the coordinates of the head will be the X and Y scalar components.  We can specify the direction of \(\vec{v}\) as

Direction Numbers(Direction Ratios) in 3D Geometry ———————————————————————————————— The term ‘Direction Numbers’ is generally introduced in 12th grade topic 3D geometry in CBSE syllabus. And here’s how it is defined.  Any three numbers which are proportional to the direction cosines of a vector are called the direction numbers of the vector.   That’s the definition of direction numbers that you will find in most indian standard textbooks, and in many online supplementary sources.  This statement is followed by expressing the same thing in mathematical language.  If a, b, c are direction numbers of a vector, then a = s cos 𝜶, b = s cos 𝜷 , c = s cos 𝜸 for any s ∈ R + If from this you are able to understand what direction numbers are then great! But in my case it didn’t really sink in at first. And the worrying part was when I found that their usage comes up almost everywhere in the topic. It took some time for me to get comfortable with them.  In this post I would like to p