Unit Vector Formula: Definition, Equations and Solved Questions

Unit Vector is a vector that has a magnitude of value 1 or unity pointing at a particular direction in space. Having unit magnitude, it may be thought to be just a direction pointed in space. Unit vectors are usually denoted by a letter with a circumflex (^) on top of it. Often, a unit vector forms the basis of a vector space and every vector in the space can be written as a linear combination of unit vectors.

Read More: Direction of Vector Formula

Key Terms: Unit Vector, Magnitude, Scalar, Vector, Direction, Coordinate System, Linear Combination, Vector space, Force, Density, Cartesian coordinate system

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What is a Unit Vector?

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A unit vector is a vector which has direction and a magnitude of 1 or unity. A vector comprises two properties namely magnitude and direction. Magnitude refers to the “size” of a vector. For a vector in a graph, its length is its magnitude, while its direction is where it is pointed or how much it is inclined to one of the axes. Some examples of vectors are velocity, force, etc. A vector that only has magnitude is a scalar. Some examples of scalars are volume, density, etc. In the case of a unit vector, the length would be one unit, while it points in the direction. Some examples of unit vectors are the basis vectors of the cartesian coordinate system. It is denoted as .

In the figure above, the blue line is a vector. It has some magnitude (||u||) and a direction is given by "theta".

In this figure, the blue and green rays are both unit vectors as their lengths are unity.

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Unit Vector Formula

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As mentioned above, a 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 write a vector: Using brackets or as a linear combination of basis vectors x, y and z.

Read More: Vector Formula

For an arbitrary vector represented as

or

Its unit vector is given as

In Bracket Form: 

In Linear Combination:

  Where,

• \(\hat{v}\) is the Unit vector of unit magnitude along the direction of vector a.
• \(\overrightarrow{v}\) is a vector with any direction and magnitude.
• \(|\overrightarrow{v}|\) is the magnitude of the vector \(\overrightarrow{v}\) .
• x, y, and z are the values of the vector in the x-axis, y-axis and z-axis respectively. 
• \(\hat{i}\) , \(\hat{j}\) , and \(\hat{k}\) are unit vectors bound for the positive axis x, y and z respectively. 

Read More: Addition of Vectors

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Calculation of a Unit Vector

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• A unit vector can be calculated from some given vector. This determines the direction pointed by the given vector. To do this, one needs both the vector expression and its magnitude.
• A vector with only magnitude and no direction is a scalar.
• Dividing the given vector by its magnitude gives the direction it is pointed at, as a unit vector. Here, the unit vector is termed the “direction vector.”
• By doing as mentioned, one reduces the size of the vector to unity only for its direction to remain.
• Conversely, multiplying a unit vector by some scalar, produces a vector, a = sâ where s is a scalar and â is a unit vector. The scalar gives magnitude to the unit vector transforming it into a regular vector.

Read More: Types of Vectors

Example:

Consider a vector a = 2i + 3j + 4k. Calculate its unit vector.

Solution: The given vector is a = 2i + 3j + 4k ------(1) To find the unit vector of some given vector,

Step 1: Find out the magnitude of the vector.  The magnitude of any vector is the square root of the sum of squares of its direction cosines. Magnitude, |a| = a = √(x2 + y2 + z2) ------- (2)

From the given vector, x = 2; y = 3 and z = 4. Substitute the values in equation (2).

Magnitude, |a| = a = √(22 + 32 + 42) = √(4 + 9 + 16) = √(29) ----------(3)

Step 2: Dividing the given vector by its magnitude. Unit vector = Vector/Magnitude Hence, â = a/|a| = (xi + yj + zk)/√(x2 + y2 + z2).

Substituting values from equation (1) and (3) â = (2i + 3j + 4k)/√(29)

Step 3: Representation

• In Bracket form, â = ((2/√(29)), (3/√(29)), (4/√(29))
• As a linear combination, â = 2i/√(29) + 3j/√(29) + 4k/√(29))

Read More: Position Vector

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Things to Remember

• Unit vectors have direction and unit magnitude.
• Unit vector = Vector/Magnitude.
• The unit vector derived from some vector points in its direction.
• Multiplying a unit vector by some magnitude produces a general vector.
• Any vector in the Cartesian Coordinate system can be represented as a linear combination of unit vectors that represent the axes of the system.
• The value of a unit vector is obtained by dividing each of the direction cosines of the vector by its magnitude.

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