Components of a vector in 2D and 3D Space

A vector is a quantity that has both magnitudes as well as direction. Resolving the components of a vector includes splitting the vectors so that the vector’s force and direction get aligned to a common axis. Resolution of the components of a vector minimises calculations. It improves the understanding of the vector with our frame of reference. The component form of a vector comprises a portion of the vector with respect to each co-ordinate system’s axis. A vector’s components can also be calculated for a vector in a three-dimensional geometric plane. Let us learn more about vector components, how to find components of a vector, and the various arithmetic operations involving vector components.

Components of a Vector in Two Dimensions

Assume V is defined in a two-dimensional plane. The vector V is divided into two parts, v_x, and v_y. Let an angle be formed between the vector V and the vector’s x-component θ. If we draw a line parallel to the vector V and its x-component (v_x), we get a right-angled triangle till (v_y).
We can deduce from trigonometric ratios that
Cos θ= Adjacent side/ hypotenuse = v_x/V
Sin θ= Opposite side/ hypotenuse = v_y/V
V= magnitude of the vector
From the above equations, we can easily say that
cos θ = v_x/V and sin θ = v_y/V
Can be written as-
v_x=V cos θ and v_y=Vsin θ
As it is a right-angled triangle, we can apply the Pythagoras theorem and get the equation for the magnitude of the vector which will be equal to-
|V| = √(vx², vy²)

Components of a Vector in Three Dimensions (Orthogonal Vectors)

Components of a vector can be easily represented in three dimensions using the co-ordinate system. Let us first define and try to understand what orthogonal representation is, before moving on to vector representation in three dimensions.
In co-ordinate geometry, orthogonal representation refers to parameters that are at right angles to one another. In an orthogonal three-dimensional system, there are three axes perpendicular to each other, representing the x, y, and z axes. Now that you know how the orthogonal system works, let us understand the concept of the unit vector.
Unit vectors are vectors with a magnitude equal to one; they are often also referred to as direction vectors, as they are mostly used to denote the direction of the vector. Unit vector can be written as-
x=x/|x|
Where,
x = unit vector
x = represents the vector
|x| = magnitude of the vector
* Plot a 3-D graph with points X(1, 1, 1), A(1, 0, 0), B(0, 1, 0) and C(0, 0, 1) on the and axes respectively.
As we can see from the figure above the magnitude of Vector OA , OB and OC is equal to one along the x,y, and z-axis respectively. These unit vectors are represented as i,j and k. With this knowledge, let us now resolve a vector in three-dimensional space.
Assume an imaginary point M with co-ordinates (x,y,z) in a 3-D space
OA = xi
OB = yj
OC = zk
The resultant vector can be represented as-
r= OM=xi+yj+zk
Magnitude of vector r by Pythagoras theorem will be obtained as-
|r|= (x²+y²+z²)

Algebraic Operations with components of a vector-

The various algebraic operations on vectors can be easily performed by utilising the vector’s various components. Consider the following two vectors:
A = a₁i+b₁j+c₁k
B = a₂i+b₂j+c₂k
Addition of these vectors will be
A + B = (a₁ +a₂) i+ (b₁+b₂) j+ (c₁+c₂) k
Subtraction of these vectors will be
A – B = (a₁ -a₂) i+ (b₁-b₂) j+ (c₁-c₂) k

Conclusion

Vectors are general mathematical representations with magnitude and direction. The components of a vector result in a vector split. The vector is divided with reference to each axis, and the components of a vector can then be computed.
The individual components of a vector can later be combined to form the vector representation as a whole. Vectors are typically represented in a two-dimensional coordinate plane with an x-axis, y-axis, or three-dimensional Space with an x-axis, y-axis, and z-axis. The vector’s direction in a two-dimensional co-ordinate system is the angle formed by the vector with the positive x-axis.