Vector in 2D and 3D space

Vectors are useful in our everyday lives. However, in the real world, events take place in three dimensions. In general, we learn to solve vectors in two dimensions. However, to enhance the usage of vectors in more realistic applications, it is necessary to explain vectors in terms of three-dimensional planes.

Let us first start with the definition of vector, In mathematics, Vector is the Quantity that has both Direction and Magnitude. For example, Force, Force has specific Quantity and direction so force can be called as vector quantity.

Now let us understand more about 2D and 3D, 

2 Dimensional which means the shape having both length and width, and 3D dimensional which means, the shape which has Length, breath, and height. 

We know that to represent a point P on a cartesian plane we write P (x, y), where x is the point on x- axis and y is the point on the y-axis, but for vectors, it is different to represent a vector on a plane.

Let us assume a vector A, shift the tail of the vector a to the origin of the coordinate system, then the head of the vector will be at some point let say (j, k) on the plane, then the point (j, k) is called as the coordinate of the vector P. with the understanding of 2D, 3D and vector, now let us understand Vector in 2D and 3D

Vector in 2D

We know that in 2D, there is only measurement in X-axis and Y-axis(Length and breadth) so in 2D vector, For Example 

Let say we have 2D vector then Vector can be written as  P  ⃗=Px + Py , this 2D vector can also be written as (Px , Py ) in rectangular form.

Where Px is the measurement of P vector in X coordinate (abscissa) and  Py is the measurement of P vector in Y coordinate (ordinate)

Addition and Subtraction of 2D vector

Adding or subtraction of 2D vectors should be done only in the same axis, if we add or subtract two vectors then, it should be done for the same component only, for example.

Let’s say we have two vectors. A= Px + Py and B= Qx + Qy 

Then the vector addition of A+B is equal to (Px + Qx ) + (Py + Qy) 

Vector in 3D

Now in 3D, We know that, there is measurement in X axis, Y axis and Z axis (Length, breadth and height) so in 3D vector, 

Let say we have 3D vector then Vector can be written as  P  ⃗=Px + Py , 

This 3D vector can also be written as (Px , Py Pz ) in rectangular form., Where Px is the measurement of P vector in X coordinate (abscissa) and  Py is the measurement of P vector in Y coordinate (ordinate).

Addition and Subtraction of 3D vector

Adding or subtraction of a 3D vector should be done only in the same axis, if we add or subtract two vectors then, it should be done for the same component only, for example.

Let’s say we have two vectors. A= Px + Py + Pz  and B= Qx + Qy + Qz

Then the vector addition of A+B is equal to (Px + Qx ) + (Py + Qy) + (Pz + Qz )

Component of 2D Vector

Any 2D vector directed in two dimensions can divide into two different directions. That is, it can have two parts. Each part of a two-dimensional vector is known as a component of 2D Vector. 

Let’s say we have two vectors. A= Px + Py ,

Then the X- component of the vector A is : Px = r.Cosθ, 

And  the Y- component of the vector A is : Py = r.Sinθ, where 

|r| = √[( Px)2+ ( Py)2]

Component of 3D Vector

Any 3D vector directed in three dimensions can divide into three different directions. That is, it can have three parts. Each part of a three-dimensional vector is known as a component of 3D Vector. 

Let’s say we have a vector. A= Px + Py + Pz  

Then the X- component of the vector A is : Px = r.Cosθ, 

And the Y- component of the vector A is : Py = r.Sinθ, 

And the Z- component of the vector A is : Pz  , 

where, |r| = √[( Px)2+ ( Py)2  + (Pz)2]

Point to Remember 

• In mathematics, Vector is the Quantity that has both Direction and Magnitude.
• Adding or subtraction of 3D vector should be done only in same axis, if we add or subtract two vectors than, it should be done for same component only