Visualizing Unit Vectors
Now to understand this let’s imagine that these are my X,Y. and Z co-ordinates. So let’s that one be my X co ordinate this one be my Y and this one be my Z. now you know that I can refer any point in space using these 3 co-ordinates. Using my box approach. Let us say I will refer to a point here itself. X, Y that be 2 ,3. So if I know the point 2,3 I have to walk 2 steps in the direction of the positive x axis and then 3 more in the direction of Y, then I reach my point. Now what are the same point representing the language of vectors why? Because we are talking about vectors, I want to talk about everything including components in the language of vectors. How do I do that? If I imagine this to be a unit vector, a vector with magnitude 1 and pointing the direction it looks like it is pointing. Then it can point infinitely many directions right? But what if I make a point in a special direction, the direction of my positive right axis. So now the tip of that is exactly at 1,0. So I want refer to 1,0, I can refer to it is 1* my vector, unit vector along X direction. I want to give it a name so I will call it X^ just to say is unit vector pointing the direction of X axis. Now I want to refer 2,0 what will I do ? I can lengthen this vector make it double its length and how will I do that by multiplying with scale right? So 2 times X^ gives me the point 2, 0. So I want to refer 2,0, I can either say 2,0 or 2 times X^. Now it must become easy to imagine what I must do for 2, 3. right? have a vectors that pointing towards my X direction unit vector. So I want to remove this pointing along Y direction. So if I take one more and say this one point along my Y direction and as you will probably guess I will call it my Y^. then the point 2,3 can be referred to its 2X^+3Y^why they are pluses? The vector plus, right? Because I am going to use my triangle law. 2X^+3Y^ which using triangle law will add up to my vector the points to the point,2,3. Now it will become easy to imagine this in 3D as well. Because all I need is to refer to any points, is now I refer to any point on this plane, that’s all I need. But if I want to refer the points in space as well, along with these two I am going to add one more,is going to point my Z axis, going to call it as Z^. now these 3 unit vectors together can refer to any point in space there by making it so that we can use a language of components within vectors. How is that? So whatever I told before right? A point X,Y,Z now becomes X X^+YY^+ZZ^. Now what I called X^, Y^ and Z^ I can also call I^ along X axis. J^ along Y axis and K^ along Z axis just another names to refer unit vectors. But why do we do this? I don’t know why do we use X^,Y^, and Z^ they really do the job well, but many many books choose to write i^,j^ and k^is just another way of saying the same thing. So now using this language if a point X,Y,Z now it will refer to as Xi^+Yj^+Zk^.