Components and Resolved Parts of a Vector

Components and Resolved Parts of a Vector

Components of a Vector

• A vector can be described by a set of components, which are effectively the vector’s coordinates.
• Coordinates or components of a vector identify its position in a space according to each dimension, for example, in a two-dimensional space - x and y-axis.
• Components of a vector can also identify its magnitude and direction in each dimension.
• A vector in two dimensions can be denoted as v = (Vx, Vy), where Vx and Vy are components along the x-axis and y-axis, respectively.
• The components of the vector are also known as its cartesian coordinates.

Resolving a Vector into Components

• Breaking a vector down into its components is known as resolving the vector.
• To resolve a vector, you may need an angle and the vector’s magnitude.

• For a vector v with magnitude

  v

  , making an angle θ with the positive x-axis in a 2D-plane, its components along x and y directions can be calculated using:

  • Vx =

    v

    cos θ

  • Vy =

    v

    sin θ

• Alternatively, if you are given components, you can calculate the magnitude

  v

  of the vector using Pythagoras’ Theorem :

  v

  = sqrt(Vx² + Vy²).

• The angle θ the vector makes with the x-axis can be found using: tan θ = Vy/Vx.

Vector Components in Three-Dimensional Space

• Vectors can also exist in a three-dimensional plane — here, they will have a third component representing the z-axis.
• A vector in three-dimensional plane can be denoted as v = (Vx, Vy, Vz).

• The magnitude in the three-dimensional plane can be calculated as

  v

  = sqrt(Vx² + Vy² + Vz²).

• Resolving vectors into their components and understanding these components is key to manipulating vectors efficiently.

Important Concepts

• Understanding vector components can help with identifying vector direction and magnitude.
• Resolving a vector into components is a useful mathematical operation in numerous fields, such as physics and engineering.
• When resolving vectors, it’s crucial to understand the angle of inclination and the unit measurements used in each dimension.
• Vector addition and subtraction can be performed easily with vector components rather than with magnitudes and directions.
• Keep in mind that resolving vectors is the reverse process of forming vectors, where the components are combined to give the vector.