Basic Properties of Vectors

There are various basic properties of vectors are:

• Components of a Vector
• Magnitude of a Vector
• Direction of a Vector

Let’s discuss these in details as follows:

Components of a Vector

Vector in a coordinate system is often decomposed into its components along the axes of that system. In a Cartesian coordinate system, a vector is broken down into its x, y, and z components (in three-dimensional space). These components describe how much of the vector lies along each axis.

In a 3D Cartesian coordinate system, a vector v can be represented as:

[Tex] \mathbf{v} = \langle v_x, v_y, v_z \rangle [/Tex]

Where v_x, v_y and v_z are the components of the vector along the x-axis, y-axis and z-axis respectively.

In three dimensions, if the angles with respect to the x, y, and z axes are α, β, and ℽ respectively

• [Tex] v_x = | \mathbf{v} | \cos \alpha [/Tex]
• [Tex] v_y = | \mathbf{v} | \cos \beta [/Tex]
• [Tex] v_z = | \mathbf{v} | \cos \gamma [/Tex]

Read More about Components of a Vector.

Magnitude of a Vector

Magnitude of a vector, often referred to as its length or norm, is a measure of how long the vector is. It is a scalar quantity that represents the distance from the vector’s initial point to its terminal point in the coordinate system.

The magnitude of a vector v, denoted as ∣v∣, is calculated using the components of the vector.

For a vector [Tex]\mathbf{v} = \langle v_x, v_y \rangle[/Tex] in a 2D Cartesian coordinate system, the magnitude is given by:

[Tex]| \mathbf{v} | = \sqrt{v_x^2 + v_y^2} [/Tex]

For a vector [Tex] \mathbf{v} = \langle v_1, v_2, \ldots, v_n \rangle[/Tex] in an n-dimensional space, the magnitude is given by:

[Tex]| \mathbf{v} | = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2} [/Tex]

Direction of a Vector

Direction of a vector specifies the orientation of the vector in space. It indicates where the vector is pointing relative to a reference direction, usually the positive x-axis in a Cartesian coordinate system.

For a vector [Tex]\mathbf{v} = \langle v_x, v_y, v_z \rangle [/Tex] in 3D space, the direction is often described using two angles:

The angle α, β, and ℽ with the x, y and z-axis respectivly.

Vectors are one of the most important concepts in mathematics. Vectors are quantities that have both magnitude and direction. A vector quantity is represented by an arrow above its head. Vectors help us understand the behaviour of directional quantities in 2D and 3D planes. Vectors are also used for determining the position and change of position of points.

Every vector follows a certain set of rules, known as the properties of vectors. It is highly important to know these properties to have a strong command of vector algebra. In this article, we will see the definition of a vector, the properties of vectors, and the properties of vector products.