# Vector

Vectors are indicated in the scientific literature with a letter, generally lowercase, with an arrow above it: $$\vec{v}$$. Vectors are essential to physics and engineering. Many fundamental physical quantities are vectors, including displacement, velocity, force, and electric and magnetic vector fields. In this context, the following fundamental entities are assigned:

• free vectors: characterized by magnitude, direction, and orientation;
• cursors: characterized by magnitude, direction, orientation, and straight-line of action;
• applied vectors: characterized by magnitude, direction, orientation, and point of application;
• versors: characterized only by direction and orientation; (the magnitude is unitary).

Let us now explain specifically what are the characteristics of a vector:

• the magnitude of a vector represents the length of the vector itself. In physics, the magnitude of a vector is identified with a number and a unit of measurement;
• the direction of a vector is the geometric straight line on which the segment lies and which also indicates the inclination with respect to the chosen reference system;
• the orientation of a vector, on the other hand, is the way toward the segment defines the vector (the arrow).

Scalar products of vectors define other fundamental scalar physical quantities, such as energy. Vector products of vectors define still other fundamental vector physical quantities, such as torque and angular momentum. In other words, vectors are a part of physics in much the same way as sentences are a part of literature. In introductory physics, vectors are Euclidean quantities that have geometric representations as arrows in one dimension (in a line), in two dimensions (in a plane), or three dimensions (in space). They can be added, subtracted, or multiplied. In this chapter, we explore elements of vector algebra for applications in mechanics and electricity and magnetism. Vector operations also have numerous generalizations in other branches of physics.

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