Vectors

You can think of a vector as an arrow that starts at the origin and reaches a point on the plane.

Every point on the Cartesian plane is reached by one and only one vector.

Therefore, you can represent a point on the plane with an ordered pair of numbers indicating the coordinates (x,y) of the point on the plane, or with a vector.

For instance, consider point P located at coordinates (2,3). The same point can be reached by vector v.

This means that the coordinates (x,y)=(4,5) and vector v represent the same point on the plane.

$$ \vec{v} = \overline{OP} = \begin{pmatrix} 4 \\ 5 \end{pmatrix} $$

Vectors are typically denoted by a letter with an arrow above it or with a bold letter.

The scalar components of a vector are the numbers that correspond to its coordinates. In this case, x=4 and y=5 are the scalar components of vector v.

Moreover, the starting point O and ending point P of the arrow are known as the endpoints of the vector. By convention, the direction of a vector is indicated by the direction of the arrow.

The point S is situated at the coordinates (5,2) and is reached by the vector w, which shares the same origin (O) as vector v but has a distinct endpoint.

Note. The origin of a vector, also referred to as the point of application, is the point at which the vector begins.

What are the components of a vector?

To fully describe a vector, we need to consider three components:

• Direction
  The line or plane in which the vector lies.
• Orientation
  The direction in which the vector points along the line or plane.
• Magnitude
  The length or size of the vector. $$ | \overline{v}| = \sqrt{x^2+y^2} $$ This is often referred to as the norm or absolute value of the vector, and is denoted by the same letter as the vector itself enclosed in vertical bars: |v|.

Note. In this example, the vector's magnitude, denoted by |v|, is equal to 6.4. This represents the length of the arrow segment on the plane. The vector is positioned on a line that indicates its direction. Meanwhile, the arrowhead points to the direction of travel along the line, which could either be in one direction or the opposite direction.

Vectors also exist in three-dimensional space.

In this case, points in space are represented by three coordinates (x, y, z).

Vectors in three dimensions are defined by three scalar components.

$$ \vec{v} = \overline{OP} = \begin{pmatrix} 2 \\ 5 \\ 4 \end{pmatrix} $$

The principles that apply to vectors on a two-dimensional plane also apply to vectors in three-dimensional space.

In this context, a vector is defined by its direction, sense, and magnitude. These three properties determine the position and orientation of the vector in space.

Note. In linear algebra, vectors can have more than three dimensions, although they may be difficult to visualize. However, for the purpose of this explanation, I will focus on vectors on the plane or in three-dimensional space as they are the easiest to understand.

What are vectors used for?

Vectors are widely used in the study of physics to describe natural phenomena. For example, gravity, motion of bodies, velocity, acceleration, etc.

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