How Do I Write a Vector?

You can write a vector in several ways, depending on the context and the level of detail needed. Here's a breakdown of common methods:

1. Using Component Form (Ordered Pairs/Triples)

This is a fundamental way to represent a vector. It expresses the vector in terms of its components along coordinate axes.

2D Vector: A vector in two dimensions is represented as an ordered pair: (x, y), where x is the horizontal component and y is the vertical component. For instance, the vector (3, 4) represents a displacement of 3 units along the x-axis and 4 units along the y-axis.
3D Vector: A vector in three dimensions is represented as an ordered triple: (x, y, z), where x, y, and z are the components along the x, y, and z axes, respectively. An example would be (1, -2, 5).

This method expresses a vector as a sum of scalar multiples of unit vectors.

Unit Vectors: Unit vectors are vectors with a magnitude (length) of 1. In a Cartesian coordinate system, the unit vectors along the x, y, and z axes are denoted as i, j, and k, respectively.
2D Vector: A 2D vector can be written as ai + bj, where a and b are scalar components along the x and y axes. For example, the vector (3, 4) can be written as 3i + 4j.
3D Vector: A 3D vector is expressed as ai + bj + ck, where a, b, and c are scalar components along the x, y, and z axes. The vector (1, -2, 5) becomes 1i - 2j + 5k (or simply i - 2j + 5k).

You can define a vector by its magnitude (length) and direction (angle).

2D Vector: You specify the magnitude r and the angle θ (theta) it makes with the positive x-axis. The components can then be found using trigonometry:
- x = r * cos(θ)
- y = r sin(θ)
  Therefore, the vector is (r*cos(θ), r*sin(θ)) or (r*cos(θ))i + (r\sin(θ))j.
3D Vector: Specifying a 3D vector using magnitude and direction is more complex, usually involving two angles (e.g., azimuth and elevation) or direction cosines.

Vectors can be written as column or row matrices, which is especially useful for linear algebra operations.

Representation	2D Example	3D Example	Description
Component Form	(3, 4)	(1, -2, 5)	Ordered pair/triple of components.
i, j, k Notation	3i + 4j	i - 2j + 5k	Sum of scalar multiples of unit vectors.
Magnitude and Direction	r = 5, θ = 53.13°	(More Complex)	Magnitude (length) and angle(s) specifying the direction.
Column Matrix	[3] [4]	[1] [-2] [5]	Matrix representation suitable for linear algebra.
Row Matrix	[3 4]	[1 -2 5]	Matrix representation suitable for linear algebra.

In essence, choosing how to write a vector depends on the specific application and the information you want to emphasize.