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Vector and Operations

Unit Vector from a Vector

Nabil Akbarazzima Fatih

Mathematics

Definition of Unit Vector

A unit vector is a vector that has a length or magnitude equal to 1 unit. Unit vectors are used to indicate the direction of a vector in space. If we have a vector, we can obtain a unit vector with the same direction by dividing the vector by its length.

Basic Concept

Consider vector PQ→\overrightarrow{PQ} in a Cartesian coordinate system. The unit vector of PQ→\overrightarrow{PQ} is defined as vector PQ→\overrightarrow{PQ} divided by its length.

a⃗PQ=PQ→∣PQ→∣\vec{a}_{PQ} = \frac{\overrightarrow{PQ}}{|\overrightarrow{PQ}|}

Where ∣PQ→∣|\overrightarrow{PQ}| is the length or magnitude of vector PQ→\overrightarrow{PQ}.

A unit vector always points in the same direction as the original vector but has its length normalized to 1 unit.

Vector and Unit Vector
Visualization of the original vector (purple) and the unit vector (yellow) that have the same direction but different lengths.

Calculating a Unit Vector

Calculation Steps

To determine the unit vector of a given vector, follow these steps:

  1. Identify the original vector
  2. Calculate the length of the vector
  3. Divide the vector by its length

Example Application

Let's say we have vector v⃗=(3,6,4)\vec{v} = (3, 6, 4)

First step, we calculate the length of the vector:

∣v⃗∣=32+62+42=9+36+16=61|\vec{v}| = \sqrt{3^2 + 6^2 + 4^2} = \sqrt{9 + 36 + 16} = \sqrt{61}

Second step, we divide the vector by its length:

v^=v⃗∣v⃗∣=(3,6,4)61\hat{v} = \frac{\vec{v}}{|\vec{v}|} = \frac{(3, 6, 4)}{\sqrt{61}}

Therefore, we get:

v^=(361,661,461)\hat{v} = \left(\frac{3}{\sqrt{61}}, \frac{6}{\sqrt{61}}, \frac{4}{\sqrt{61}}\right)
Example of Unit Vector in 3D Space
Visualization of vector v(3,6,4) and its unit vector v̂ pointing in the same direction.

Properties of Unit Vectors

Length of a Unit Vector

A unit vector always has a length equal to 1. This can be proven by calculating the length of the unit vector:

∣v^∣=(361)2+(661)2+(461)2=9+36+1661=6161=1|\hat{v}| = \sqrt{\left(\frac{3}{\sqrt{61}}\right)^2 + \left(\frac{6}{\sqrt{61}}\right)^2 + \left(\frac{4}{\sqrt{61}}\right)^2} = \sqrt{\frac{9 + 36 + 16}{61}} = \sqrt{\frac{61}{61}} = 1

Unit Vectors on Coordinate Axes

In a three-dimensional coordinate system, there are three standard unit vectors that are parallel to each coordinate axis:

  • ı^\hat{\imath} is the unit vector in the direction of the x-axis
  • ȷ^\hat{\jmath} is the unit vector in the direction of the y-axis
  • k^\hat{k} is the unit vector in the direction of the z-axis

Any vector can be expressed as a linear combination of these three unit vectors.

Standard Unit Vectors
Standard unit vectors on the coordinate axes.

Applications of Unit Vectors

Indicating Direction

Unit vectors are very useful for indicating direction without regard to magnitude or length. In physics, for example, unit vectors are often used to indicate the direction of force, velocity, or acceleration.

Physics Calculations

In physics, when we want to decompose a vector into its components, unit vectors are very helpful. For instance, a force can be decomposed into components along the x, y, and z axes using unit vectors.

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