Vector Length

In mathematics, the length of a vector in Euclidean space refers to the distance from the origin to the point represented by the vector. This concept is also known as the magnitude of a vector. The length of a vector is always a non-negative real number or zero. It is denoted by $\Vert \mathbf{v}\Vert$, where $\mathbf{v}$ is the vector.

Euclidean Space

Euclidean space is a geometric space that is flat and extends infinitely in all directions, obeying the Euclidean geometry laws introduced by the ancient Greek mathematician Euclid. In Euclidean space, vectors are represented by directed line segments with a magnitude and a direction.

Calculating the Length of a Vector

The length of a vector $\mathbf{v}=(v_1,v_2,...,v_n)$ in $n$-dimensional Euclidean space can be calculated using the formula:

$\Vert \mathbf{v}\Vert =\sqrt{v_1^2+v_2^2+...+v_n^2}$

This formula is derived from the Pythagorean theorem, where the length of each component of the vector is squared, summed, and then the square root is taken.

Examples

Example 1

Consider a 2-dimensional vector $\mathbf{v}=(3,4)$. The length of $\mathbf{v}$ can be calculated as:

$\Vert \mathbf{v}\Vert =\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5$

Therefore, the length of $\mathbf{v}$ is 5.

Example 2

Let $\mathbf{u}=(1,-2,2)$. The length of $\mathbf{u}$ is:

$\Vert \mathbf{u}\Vert =\sqrt{1^2+(-2{)}^2+2^2}=\sqrt{1+4+4}=\sqrt{9}=3$

Hence, the length of $\mathbf{u}$ is 3.

Real-life Example

Imagine you are standing in a field. You decide to walk 3 units east and 4 units north. The motion you make can be represented as a vector $(3,4)$. The length of this vector represents the straight-line distance from where you started to where you ended up after moving east 3 units and north 4 units.

Historical Context

The concept of vector length/magnitude dates back to the early days of Cartesian geometry when Descartes introduced the idea of representing geometric entities with ordered pairs of real numbers. The notion of vector length was further formalized and generalized in the modern mathematical framework to encompass vectors in higher dimensions.

Exam Questions

1. Calculate the length of the vector $\mathbf{w}=(2,-1,3)$.
2. In a 3-dimensional space, if the vectors $\mathbf{a}=(1,2,-2)$ and $\mathbf{b}=(3,-1,4)$ are added together, find the length of the resulting vector.
3. True or false: The length of a vector can be negative.

Remember that the length of a vector provides valuable geometric insight into the magnitude of the vector in Euclidean space and is a fundamental concept in various mathematical disciplines.