Unit Vectors: Definition, Formula and Examples

A vector is a quantity having both magnitude as well as direction. A unit vector is a vector with a magnitude of 1. Unit vectors are often denoted by placing a hat symbol (ˆ) above the vector variable.

• Unit vectors are also referred to as the direction vector.
• The unit vector plays an important role in vector analysis, physics, and computer graphics.
• They are used to describe directions without being affected by the scale of the vector.
• The unit vector of the vector by simply dividing the vector by its magnitude. 
• For a non-zero vector v in a vector space, the unit vector v in the same direction is calculated by dividing each component of v by its magnitude (∣v∣).

\(\hat{v} = {v \over |v|}\)

Key Terms: Unit Vector, Vectors, Magnitude, Length, Direction Vector, Vector Analysis, Variable, Length, perpendicular vector

Read More: Co Planar Vectors

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What is Unit Vector?

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A unit vector represents a vector that has a magnitude of 1. Unlike regular vectors that can vary in length, unit vectors maintain the direction of the original vector while having a fixed length of 1.

Unit Vector Definition: A unit vector is calculated by dividing each vector component by its magnitude.

• It is often denoted by a hat symbol (ˆ) on top of a vector variable
• The defining characteristic of a unit vector is its magnitude, which is always 1. 

Unit Vector

Unit Normal Vector

The normal vector is perpendicular to a surface at a given point. It is also known as the "normal," it is a vector pointing away or towards the surface.

• On closed surfaces, two types of normals are found - pointing towards the interior and pointing outward.
• Obtained by normalizing a nonzero normal vector, it is sometimes called the "unit normal."

So, the normal vector is a perpendicular vector to a surface, and its unit form is the unit normal vector achieved by normalization.

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Magnitude of Unit Vector

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The magnitude of a vector, often denoted by |v∣ represents the length or size of the vector in a geometric sense. In two-dimensional space, the magnitude of a vector v = ⟨x,y⟩ is calculated using the Pythagorean theorem. In three-dimensional space, the magnitude is calculated, considering three components.

Mathematically, for a 2D vector: ∣v∣ = \(\sqrt {x^2 + y^2}\)​

For a 3D vector: \(|v| = \sqrt {x^2 + y^2 + z^2}\)

The magnitude of a vector is always a non-negative value. It provides information about the vector's length but not its direction.

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Unit Vector Symbol

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The symbol for a unit vector is often represented by placing a hat (ˆ) or a circumflex accent (^) above the vector symbol. 

For example: If v = ⟨3,4⟩ is a 2D vector, then the corresponding unit vector is \(\hat{v} = {⟨3,4⟩\over∣⟨3,4⟩∣}\)

The hat symbol indicates that the vector has been normalized to have a magnitude of 1, turning it into a unit vector. The unit vector points in the same direction as the original vector but has a length of 1. 

Read More: Determinant Formula

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Unit Vector Formula

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The unit vector, often denoted as \(\hat{v}\), for a given vector v, is obtained by dividing each component of the vector by its magnitude (∣v∣).

• In mathematical terms, the formula for the unit vector \(\hat{v} = {v \over |v|}\)
• Here, v is the original vector, and ∣v∣ represents the magnitude of the vector.
• It can also be represented as \(\begin{array}{l}\vec{a}= x\hat{i} + y \hat{j} +z \hat{k}\end{array}\)

Magnitude of a vector Formula: \(\begin{array}{l}|\vec{a}|= \sqrt {x^2 + y^2 +z^2}\end{array}\)

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Unit Vector Solved Example

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The solved examples on unit vector formula are as follows – 

Example 1: Vector Given: \((\vec{v} = 3i - 4j + 5k)\)

Solution: Step 1:

Given Vector, \((\vec{v} = 3i - 4j + 5k)\)

(x-component of the vector (a)) = 3

(y-component of the vector (b)) = -4

(z-component of the vector (c)) = 5

Step 2: Magnitude of Vector \((v) = (|\vec{v}| = \sqrt{3^2 + (-4)^2 + 5^2} = \sqrt{9 + 16 + 25} = \sqrt{50})\)

Step 3: \((\vec{v} = \frac{3i - 4j + 5k}{\sqrt{50}})\)

\(\hat{v} = \left(\frac{3}{\sqrt{50}}i - \frac{4}{\sqrt{50}}j + \frac{5}{\sqrt{50}}k\right)\)

This is the required unit vector.

Example 2: Find the unit vector of \((\vec{u} = -i + 2j - 2k)\)

Solution: Step 1: Given Vector, \((\vec{u} = -i + 2j - 2k)\)

x-component of the vector (a) = -1

y-component of the vector (b) = 2

z-component of the vector (c) = -2

Step 2: Magnitude of Vector (u) = \((|\vec{u}| = \sqrt{(-1)^2 + 2^2 + (-2)^2} = \sqrt{1 + 4 + 4} = \sqrt{9})\)

Step 3: \((\vec{u} = \frac{-i + 2j - 2k}{\sqrt{9}})\)

\(\hat{u} = \left(\frac{-1}{3}i + \frac{2}{3}j - \frac{2}{3}k\right)\)

This is the required unit vector.

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Application of Unit Vector

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The applications of unit vectors are as follows:

1. Unit vectors denote direction in physics, especially in representing velocity vectors.
2. They are essential in force analysis, breaking forces into components for problem-solving.
3. Unit vectors are used to express electric and magnetic fields in electromagnetism.
4. They are commonly used in computer graphics for defining light, camera orientation, and object transformations.
5. Unit vectors are used in structural analysis to break down complex forces into simpler components.
6. They are used to represent the state of a quantum system in quantum mechanics.

Types of Vectors Detailed Video Explanation:

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Things to Remember

1. Unit vectors have a magnitude of 1 and are used to represent direction in vector spaces.
2. Normalized Form is obtained by dividing a vector by its magnitude, resulting in a unit vector pointing in the same direction.
3. Denoted by a caret symbol, such as \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\), representing unit vectors in the x, y, and z directions.
4. They facilitate vector operations and calculations due to their normalized form.
5. Unit vectors are widely used in physics, engineering, computer graphics, and mathematics for direction representation.
6. Normalization Formula is achieved by dividing each component of a vector by its magnitude, ensuring it becomes a unit vector.