Vector Components in 2D and 3D Space

A vector consists of components as per the direction in which those are functioning. The direction of vectors can be taken as the axis in direction of which the vector is functioning. 2D vectors have only two components to them, and hence can be plotted in an XY plane. However, the 3D vectors have three components. One other direction is to be taken into account if we are to compute the magnitude and direction of that entire vector.

Components of vectors

The components of a vector help in the division of that vector into segments as per the direction of the plane in which they exist. To perform advanced arithmetic operations, vectors are needed to be split into their components to carry out further calculations. The component of a vector gives us the reference about the direction of the axis in which the vector currently exists.

It is most common to presume a vector to be a 3D geometrical entity. Seeing as a vector in its entirety gives an account of the relationship that exists between the x-axis, y-axis and the x-axis for a certain fixed point. Now, using the direction of that point on each axis, it gets easier for us to compute its properties. And, that is where the notion of components of a vector comes from.

The vectors exist in a two-dimensional coordination plane, within a three-dimensional space. This space is the one that comprises the x-axis, y-axis and the x-axis. The direction of that vector system is given as the angle that is made by the vector with the positive direction of the x-axis. 

We can divide a vector into two segments: its magnitude and its direction. These both require specific methods to bring that into effect. 

What is the difference between 2D and 3D vectors?

A vector can be termed a 2D vector when the extent of its magnitude and direction is completely known to us. In this case, we are aware of the fact that the vector is only able to go only to a finite distance from the origin of the plane. The same property can be measured for each direction of that vector. 

The magnitude for a 2D vector is represented as the longest side or hypotenuse of a triangle. While the other two sides are formed by the x-axis and the y-axis measure for that vector. The magnitude of the vector will be equal to the largest side of the triangle that is formed (hypotenuse). 

We can always find the magnitude of a 2D vector by using the Pythagoras theorem in it. As the triangle formed in the case of 2D vectors is always right-angled. And, the measure of angle for that triangle is also possible using the tangent function for that triangle. 

The 3D vectors are very similar to 2D vectors. But in this case, there is one more direction that is to be taken into account. The three components of these vectors represent a different direction for each case. In the 3D vectors, ‘i’ is used to represent the ‘x’ component of the vector, ‘j’ is used to represent the ‘y’ component of the vector, and ‘k’ is used to represent the ‘z’ component of the vector. 

Components of vector formula

Magnitude

For a 2D vector: a = (a1, a2)

|a| = √[(a1)2 + (a2)2]

For a 3D vector: a = (a1, a2, a3)

|a| = √[(a1)2 + (a2)2 + (a3)2]

The angle of a vector

The angle between two vectors is given by the following formula:

θ = cos-1 [ (a · b) / (|a| |b|) ]

Triangle law of addition for a vector

This method is greatly applicable for the 2D vectors. The result of this method is as follows:

R→ = A→ + B→

Parallelogram law of vector addition

This vector is quite similar to the triangle law of vector addition:

R→ = A→ + B→

Subtraction of vectors

R→ = A→ – B→

Conclusion

The components of a vector are the direction of the axis along which they are functioning. ‘i’, ‘j’ and ‘k’ are used to represent the ‘x’, ‘y’, and ‘z’ components of that particular vector. This fact stands true when the given vector is a 3D one. But when the vector is a 2D vector it can be measured only by two of these three. And, it can be drawn only in two planes. For a 3D vector, there must be another axis to fully explain its existence.