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Scalars and vectors are mathematical objects that are used to quantify physical quantities.

$x_1<x_2<\cdots<x_{k}$

$x_1>x_2$

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Scalar. A scalar is a real number. For example $2.23$ , $0$ , $-1324$ , $\pi$ , and so on .

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$x_1>x_2$

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Scalar quantity. A scalar quantity is a quantity that can be described by a single real number. This number specifies the magnitude or size of that quantity. For example length, angle, mass, speed, area, temperature, and pressure are scalar quantities.

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Sometimes a scalar quantity is simply referred to as scalar. For example, “mass is a scalar” is equivalent to stating: “mass is a scalar quantity”.

The mathematical operations on scalars follow the usual rules of arithmetic.

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Vector. A vector is a mathematical object that has a size (magnitude) and a direction. This definition perfectly fulfills the requirements of engineering mechanics. However, for a more rigorous mathematical definition, please refer to section 2.8.

Vector quantity. A vector quantity is a quantity characterized by both a magnitude and a direction. For example velocity, force, acceleration, and moment are all vector quantities.

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The magnitude (sometimes referred to as the norm) of a vector is a positive scalar. For example the magnitude of the velocity of a moving object is the speed of that object. Speed is always measured and reported as a positive number. The magnitude of a physical quantity must have a unit.

The line that is collinear with an arrow is called the line of action. The orientation or direction of an arrow (its tail is already known or defined) with respect to a fixed axis is defined by an angle formed between the axis and the line of action of the arrow. Fig. 2.2 shows a directed line segment or arrow representing the vector $\bold F$ with its defined terms.

Figure 1. Geometric representation of a directed line segment (arrow) or a vector. — Fig. 2.2 Geometric representation of a directed line segment (arrow) or a vector.

Zero vector. The zero vector, denoted by $\bold 0$ or $\overrightarrow 0$ , is a vector of length zero. The zero vector does not point toward any direction, therefore, its direction is undefined.

It should be noted that a vector (directed line segment) is defined by only two parameters being magnitude and direction. Therefore, the position of a vector in the space does not change the properties. As a result, two vectors are equal if they have the same direction and magnitude. Examples of equal vectors are shown in Fig. 2.3a.

Parallel vectors. Two vectors are parallel if they have parallel lines of action. In other words, two parallel vectors have the same or opposite sense of directions. Examples of parallel vectors are shown in Fig. 2.3.b.

Fig. 2.3 (a) equal vectors, (b) parallel vectors

This geometric representation applies to vectors in a plane (a two-dimensional space) or in a three-dimensional space.