Vector

What is a Vector?

Vectors are fundamental in mathematics and physics, providing a way to represent quantities that possess both magnitude and direction. Magnitude is the size or length of the vector and represents the quantity’s amount or value. For example, in the case of force, the magnitude would be the strength of the force.

Vectors also specify the direction in which the quantity is acting. For instance, a velocity vector indicates how fast an object is moving and the direction of its motion.

Components of a Vector

Magnitude

The magnitude of a vector quantifies its size or length and measures the vector’s overall effect regardless of its direction. This magnitude is usually represented by $\Vert v\Vert$ or $\vert v\vert$ and can be calculated using mathematical formulas from the vector’s components.

For a vector $v$ in two-dimensional space, with components $v_x$ (along the x-axis) and $v_y$ (along the y-axis), the magnitude is found using the Pythagorean theorem. This theorem relates the lengths of the sides of a right triangle, which equates to the vector’s components and magnitude in the context of vectors. The formula for calculating the magnitude $\Vert v\Vert$ of a two-dimensional vector is:

$\Vert v\Vert=\sqrt{v_x^2 + v_y^2}$

In three-dimensional space, a vector $v$ has three components: $v_x$ , $v_y$ , and $v_z$ . The magnitude of this three-dimensional vector is calculated using the Euclidean norm formula, which is an extension of the Pythagorean theorem to three dimensions:

$\Vert v\Vert=\sqrt{v_x^2 + v_y^2 + v_z^2}$

Direction

The direction of a vector is a crucial component that distinguishes it from a scalar quantity. It gives information about the vector’s orientation in space relative to a reference axis or plane. Here’s how direction is understood and represented in vector analysis:

Angle Measurement: The direction of a vector is often described by its angle with a reference axis, typically the positive x-axis in a two-dimensional coordinate system. This angle is usually measured counterclockwise from the reference axis, a convention that helps consistently define the direction of vectors.
Unit Vectors: Unit vectors are used to denote direction specifically. A unit vector has a magnitude of 1 and points in a particular direction. In three-dimensional space, the standard unit vectors are:
- i, which points along the positive x-axis,
- j, which points along the positive y-axis, and
- k, which points along the positive z-axis.

Vector Representation

Vectors are typically represented graphically as arrows. The length of the arrow corresponds to the magnitude of the vector, and the direction of the arrow indicates the direction of the vector quantity.
In mathematical notation, vectors are often denoted using boldface letters (e.g., $\bold v$ ) or with an arrow symbol ( $\vec{v}$ ) above the letter.