Unit Vector Formula: Definition, Example, Notation and Calculation

Unit Vector Formula Overview

A unit vector is a vector with a magnitude of 1. It plays a crucial role in many branches of physics, mathematics, engineering, and computer science. The concept of unit vectors is fundamental to understanding various physical quantities such as force, velocity, and acceleration. The unit vector formula is a mathematical expression that helps us calculate the unit vector of a given vector. In this article, we will discuss the unit vector formula, its derivation, properties, and applications.

What is Unit Vector?

A unit vector is a vector with a magnitude of 1 and is often used to indicate the direction of a vector without regard to its length or magnitude. The unit vector is found by dividing a given vector by its magnitude. The resulting vector will have the same direction as the original vector but will have a magnitude of 1. The unit vector is represented by placing a caret (^) symbol above the vector, such as v⃗^ or u⃗^. The unit vector is important in vector calculus and physics because it allows us to simplify calculations and analysis involving vectors by reducing the vectors to their direction only.

Unit vectors are also used to represent the direction of a force, velocity, or acceleration in physics. In addition, they are used to decompose vectors into their components along the x, y, and z axes, which is useful for analyzing and manipulating vectors in three-dimensional space.

Unit Vector Symbol

The unit vector symbol is represented by a caret (^) placed on top of a vector, such as v⃗^ or u⃗^. The symbol represents a unit vector, which is a vector with a magnitude of 1 that is in the same direction as the original vector.

The unit vector symbol is used in vector calculus and physics to represent a vector's direction without considering its magnitude. The direction of a vector is important in many applications, such as in determining the direction of a force or the orientation of an object. Using the unit vector symbol allows us to simplify calculations and analysis involving vectors by reducing the vectors to their direction only.

The unit vector symbol is also used to indicate a vector component in a particular direction, such as i^, j^, or k^, which represent the unit vectors along the x, y, and z axes in three-dimensional space. These unit vectors are often used to decompose vectors into their components along the x, y, and z axes, making it easier to analyze and manipulate them.

Unit Vector Formula

The unit vector is a vector with direction and unit magnitude. Consequently, a unit vector can be expressed as a division of a vector by its magnitude.

Unit Vector = (Vector)/Magnitude

The magnitude of a vector is:

There are two ways to show a vector: 

• Using brackets 
• Linear combination of basis vectors x, y and z.

The arbitrary vector can be represented as:

Or 

Its unit vector is given as

In Bracket Form: 

In Linear Combination:

 Where,

• v^ is a unit vector with a magnitude of 1 that points in a specific direction.
• v￫is a vector that can have any direction and magnitude.
• |v￫| denotes the magnitude (or length) of the vector v￫.
• x, y, and z represent the values (or components) of the vector v￫along the x, y, and z axes, respectively.
•  i^, j^, and k^ are unit vectors that point in the positive x, y, and z directions, respectively.

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Derivation of Unit Vector Formula

The unit vector formula can be derived using the Pythagorean theorem and the definition of a unit vector. Consider a vector A in two-dimensional space, given by:

A = ai + bj

where a and b are the x and y components of the vector A, respectively. The magnitude of A can be calculated using the Pythagorean theorem as follows:

|A| = √(a² + b²)

To obtain the unit vector Â, we need to divide A by its magnitude, i.e.,  

 = A/|A|

Substituting the value of A, we get:

 = (ai + bj)/√(a² + b²)

Multiplying the numerator and denominator by √(a² + b²), we get:

 = (ai + bj)√(a² + b²)/(a² + b²)

Simplifying this expression, we get:

 = (a/√(a² + b²))i + (b/√(a² + b2))j

This is the unit vector formula for a two-dimensional vector A. The formula can be extended to three-dimensional vectors by adding a k component, where k represents the z-axis.

Properties of Unit Vector Formula

The unit vector formula has several properties that are important to understand. These properties are:

1. A unit vector has a magnitude of 1.
2. A unit vector has the same direction as its parent vector.
3. The sum of two-unit vectors is also a unit vector.
4. The dot product of two-unit vectors is equal to the cosine of the angle between them.
5. The cross product of two-unit vectors is also a unit vector, which is perpendicular to both vectors.

Read more about the Orbital Velocity Formula.

Calculations of Unit Vector Formula

1. A unit vector can be calculated from a specified vector. The specified vector's direction is determined by this. This necessitates the use of both the vector expression and its size.
2. A scalar vector is one that has only magnitude and no orientation.
3. A unit vector is obtained by dividing the provided vector by its magnitude. In this case, the unit vector is referred to as the "direction vector."
4. By doing so, one lowers the size of the vector to unity while keeping its direction unchanged.
5. In the opposite case, multiplying a unit vector by a scalar yields the vector a = sâ, where s is a scalar and â is a unit vector. The scalar transforms the unit vector into a normal vector by adding magnitude to it.

Applications of Unit Vector Formula

The unit vector formula has many applications in various fields, including physics, engineering, computer science, and mathematics. Here are some examples:

1. Force and Motion: The unit vector formula is used to calculate the direction of forces and motion. For example, in projectile motion, the unit vector is used to determine the initial velocity and acceleration of the projectile.
2. Geometry and Trigonometry: The unit vector formula is used to represent and transform vectors in 2D and 3D spaces. It can be used to solve problems related to angles, distances, and positions in geometry and trigonometry.
3. Computer Graphics and Animation: The unit vector formula is used to create and animate 3D objects and scenes in computer graphics and animation. It is used to calculate the direction of light rays, camera movements, and object transformations.
4. Robotics and Control Systems: The unit vector formula is used to represent and control the motion and orientation of robots and machines. It is used to calculate the direction of movement, torque, and force in robotic systems.
5. Electromagnetism and Waves: The unit vector formula is used to calculate the direction and polarization of electromagnetic waves. It is used to solve problems related to antennas, radar, and communication systems.

Solved Examples of Unit Vector Formula

Example 1: Find the unit vector in the direction of the vector v⃗= -2i^ + 4j^ + k^.

Solution:To find the unit vector, we first need to calculate the magnitude of the vector:

|v⃗| = sqrt ((-2)^2 + (4)^2 + (1)^2) = sqrt(21)

Next, we can apply the unit vector formula:

u = v⃗/ |v⃗| = (-2i^ + 4j^ + k^) / sqrt (21)

Simplifying, we get:

u = (-2/ sqrt (21))i^ + (4/ sqrt(21))j^ + (1/ sqrt(21))k^

Therefore, the unit vector in the direction of v⃗is (-2/ sqrt (21))i^ + (4/ sqrt(21))j^ + (1/ sqrt(21))k^.

Example 2: Find the unit vector that is perpendicular to both the vectors u⃗= i^ - 2j^ + 3k^ and v⃗= 2i^ + 4j^ - 6k^.

Solution:We can find the cross product of the two vectors to obtain a vector that is perpendicular to both:

u⃗× v⃗= (i^ - 2j^ + 3k^) × (2i^ + 4j^ - 6k^) = -20i^ + 12j^ + 8k^

Next, we can calculate the magnitude of this vector:

|u⃗× v⃗| = sqrt ((-20)^2 + (12)^2 + (8)^2) = 4sqrt(21)

Finally, we can use the unit vector formula to obtain the unit vector:

u = (u⃗× v⃗) / |u⃗× v⃗| = (-20/4sqrt (21))i^ + (12/4sqrt(21))j^ + (8/4sqrt(21))k^ = (-5/sqrt(21))i^ + (3/sqrt(21))j^ + (2/sqrt(21))k^

Therefore, the unit vector that is perpendicular to both uand v⃗is (-5/sqrt (21))i^ + (3/sqrt (21))j^ + (2/sqrt (21))k^.

Example 3: Given a unit vector u = (1/3)i^ + (2/3)j^ + (2/3)k^, find the vector v⃗that has a magnitude of 5 and is parallel to u.

Solution:Since the vector v⃗is parallel to u, we can express it as a scalar multiple of u:

v⃗= cu

where c is a scalar. We want the magnitude of v⃗to be 5, so we can set up an equation:

|v⃗| = |cu| = |c||u| = 5

Since u is a unit vector, |u| = 1. Therefore, we can solve for c:

|c| = 5

We take the positive value of c since we want v⃗to point in the same direction as u. Thus, c = 5, and the vector v⃗is:

v⃗= 5u

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Things to Remember: Unit Vector Formula

• Unit vectors have a direction and a value.
• Vector/magnitude = unit vector.
• The unit vector derived from some vector points in its path.
• A general vector is created by multiplying a unit vector by some number.
• Any vector in the Cartesian Coordinate system can be expressed as a linear combination of unit vectors representing the system's axes.
• A unit vector's value is calculated by splitting its direction cosines by its magnitude.