Unit Vector

Unit Vector: Vector quantities are physical quantities that have both magnitude and direction. We define them using their initial point and final point. Unit vectors are the vectors that have a magnitude of 1. For example \vec A    = 2i + 3j is not a unit vector as its magnitude is,

|A| = √(2² + 3²) = √(13) ≠ 1

Unit vectors are the vectors that are used to give the direction of the vector. We can easily get the unit vector of the vector by simply dividing the vector by its magnitude.

In this article, we will learn about what is a unit vector, its formula, examples, and others in detail.

What is a Unit Vector?

The unit vector of any given vector is the vector obtained by dividing the given vector by its magnitude. As the name suggests a unit vector is a vector whose magnitude is one(1). It is depicted by any English letter with an inverted V or cap on top of it. Unit vectors have a length of one. Unit vectors are commonly used to indicate a vector's direction. The direction of a unit vector is the same as that of the provided vector, but its magnitude is one unit.

Unit Vector Definition

A unit vector is a vector with a magnitude of 1. It is used to specify a direction and has no other magnitude

We define a unit vector in each 3-D axis as,

• Unit vector in the x-direction is i
• Unit vector in the y-direction is j
• Unit vector in the z-direction is k

Also, the magnitude of this vector is,

• |i| = 1
• |j| = 1
• |k| = 1

The unit vector in the x, y, and z direction unit vectors are shown in the image below:

Before learning more about them let's first learn about the magnitude of the vector.

Magnitude of Vector

The strength of any vector is given by the magnitude of the vector. Any vector has both magnitude and direction and the magnitude of the vector is calculated by taking the sum of the square of individual components of the vector and then taking its square root. For any \vec{A}    = ai + bj + ck the vector's magnitude formula is,

|\vec{A}|= √(a² + b² + c²)

We can understand this concept with the help of the example discussed below:

Example: Find the magnitude of vector \vec{V} = 2i + 3k - j

Solution:

Given Vector,

\vec{V}    = 2i + 3k - j

Magnitude of vector is:

|V| = √(2² + 3² + (-1)²)

⇒ |V| = √(14) units

Unit Vector Notation

As we know vectors are represented by taking the Arrow above the number or symbol. In the same way, Unit Vectors are represented by the symbol ‘^’, which is called a cap or hat, such as \hat{a} .

And the value of the unit vector is calculated using the formula,

\bold{\hat{a} = \vec{a}/|a|}

Where |a| represents the magnitude of vector a.

Unit Vector in Three-Dimension

We measure any quantity in the three dimensions and hence the vector is also measured in 3-D space. The 3-D space is divided into three dimensions namely the x-axis, y-axis, and z-axis. These three dimensions dived the 3-D space into eight coordinates. Thus, the unit vector in each dimension is denoted as,

• Unit vector in the x-direction is i and |i| = 1
• Unit vector in the y-direction is j and |j| = 1
• Unit vector in the z-direction is k and |k| = 1

The dot product of these unit vectors is represented as,

• i.i = j.j = k.k = 1 {Since dot product is given as ab cosθ and for same unit vectors θ = 0. Hence Cos 0 = 1}
• i.j = j.k = k.i = 0 {Since dot product is given as ab cosθ and for two different unit vectors θ = 90. Hence Cos 90 = 0}

Unit Normal Vector

Any vector perpendicular to the surface at any point is called the Normal Vector to that surface at that point. And the unit vector of that normal vector is called the Unit Normal Vector and it is used to tell the direction of the surface at any particular point.

Unit Vector Formula

The formula to calculate the unit vector is,

\bold{\hat{v}=\frac{\vec v}{|\vec v|}}

Where denotes Vector ai + bj + ck and |\vec v|    denotes the Magnitude of Vector [|\vec{v}| = \sqrt{a^2 + b^2 + c^2}]

As we know that unit vectors along any vector are calculated by taking the ratio of the vector along with its magnitude. So it is very important to find the magnitude of the vector first. Any vector can be represented in two ways,

• \vec{A}   = (a, b, c)
• \vec{A}   = ai + bj + ck

And now we can easily calculate the magnitude of this vector as,

|A| = √(a² + b² + c²)

Now finding the unit vector using the unit vector formula as

Unit Vector = Vector / Magnitude of Vector

The image added below shows the image of the unit vector formula.

How to Calculate the Unit Vector?

We can easily calculate the unit vector of any given vector by following the steps discussed below:

Step 1: Write the given vector and note its component in x, y, and z directions respectively.

Step 2: Find the magnitude of the vector using the formula,

|A| = √(a² + b² + c²)

Step 3: Find the unit vector by using the formula,

Unit Vector = Vector / Magnitude of Vector

Step 4: Simplify to get the required unit vector.

We can better understand this with the help of an example,

Example: Find the unit vector of  \vec{a}   = 2i + j + 2k

Solution:

Step 1:

Given Vector,

\vec{a}   = 2i + j + 2k

• x-component of the vector (a) = 2
• y-component of the vector (b) = 1
• z-component of the vector (c) = 2

Step 2:

Magnitude of Vector (a) = |a| = √(2² +1² +2²) = √(9) = 3

Step 3:

\vec{a}   = (2i + j + 2k)/3

\hat{a}  = (2/3i + 1/3j + 2/3k)

This is the required unit vector.

How to Represent Vector in Bracket Format?

For any vector given as,

 \vec{a}   = (x, y, z)

Its unit vector is calculated and represented as,

\hat{a} = \vec{a}/|a|

           = (x, y, z)/(√x² + y² + z²)

           = [x/(√x² + y² + z²), y/(√x² + y² + z²), z/(√x² + y² + z²)]

How to Represent Vector in Unit Vector Component Format?

For any vector given as,

\vec{a}   = xi + yj + zk

Its unit vector is calculated and represented as, \hat{a} = \vec{a}/|a|

           = (xi + yj + zk)/(√x² + y² + z²)

           = [x/(√x² + y² + z²)i + y/(√x² + y² + z²)j + z/(√x² + y² + z²)k]

where i, j, and k represent the unit vector in the x, y, and z directions respectively.

Unit Vector Parallel to Another Vector

To find a unit vector that is parallel to another vector v, you need to normalize v. This is done by dividing the vector by its magnitude. Mathematically, the unit vector v^ that is parallel to v can be calculated using:

v^ = v/∣v∣​

Where ∣v∣ is the magnitude of v. This results in a vector v^ that has the same direction as v but a magnitude of 1.

Unit Vector Perpendicular to Another Vector

To determine a unit vector that is perpendicular to another vector, you need to start with a vector that is orthogonal (perpendicular) to the original vector and then normalize it. In three dimensions.

For example, if you are given a vector v=(v_x​ ,v_y ,v_z​), a perpendicular vector can be obtained through a cross product with another non-parallel vector (commonly a standard basis vector). Once you have a perpendicular vector w, you can then normalize it to find the unit vector:

w^=w/∣w∣​

where ∣w∣ is the magnitude of w. The resulting vector w^ will be perpendicular to v and have a magnitude of 1.

Applications of Unit Vector

Unit vectors have various applications. It is used in explaining various concepts of both Physics and Mathematics. Some of the common applications of unit vectors are,

• Unit vectors are used to give the direction of vectors in 2-D or 3-D planes.
• Unit vectors are responsible for representing the vectors easily in 2-D or 3-D planes.
• Unit vectors give us the output of all the forces acting on any object.
• In electromagnetism, electrostatics, and mechanics unit vectors represent various quantities and their directions which are very helpful in the study of various concepts in sciences.
• Unit vectors are used to trace the path of Missiles, Aeroplanes, Satellites, and others.

Properties of Vectors

Various properties that are useful in the study of vectors and that help us to solve various problems including the vector are,

• Dot product of two vectors is commutative, i.e. A . B = B. A
• Cross product of two vectors is not commutative, i.e. A × B ≠ B × A
• Dot product of two vectors represents a scalar quantity in the plane of two vectors
• Cross product of two vectors represents a vector quantity, perpendicular to the plane containing the two vectors.
• i . i = j . j = k . k = 1
• i . j = j . k = k . i = 0
• i × i = j × j = k × k = 0
• i × j = k; j × k = i; k × i = j
• j × i = -k; k × j = -i; i × k = -j

People Also Read,

• Scalar and Vector
• Vector Operations
• Resultant Vector Formula

Unit Vector Examples

Example 1: Find the unit vector of 2i + 4j + 5k.

Solution:

Given Vector,

v = 2i + 4j + 5k

Magnitude of Vector v

= |\vec v| = \sqrt{a^2 + b^2 + c^2} \\=  \sqrt{2^2 + 4^2 + 5^2}\\=\sqrt{45}

= 3√5

Unit Vector of v

\hat{V}=\frac{2i + 4j + 5k}{3√5} 

           = (2/3√5) i + (4/3√5) j + (√5/3)k

Example 2: Find the unit vector of 3i + 4j + 5k.

Solution:

Given Vector,

v = 3i + 4j + 5k

Magnitude of Vector v

= |\vec v| = \sqrt{a^2 + b^2 + c^2} \\=  \sqrt{3^2 + 4^2 + 5^2}\\=\sqrt{50}

= 5√2

Unit Vector of v

\hat{v}=\frac{3i + 4j + 5k}{5√2}

          = (3/5√2) i + (4/5√2) j + (1/√2)k

Example 3: Find the unit vector of the resultant of vector i + 3j + 5k and -j - 3k.

Solution:

Given Vectors,

A =  i + 3j + 5k

B =  -j - 3k

Resultant Vector = R = A + B

= ( i + 3j + 5k) + (-j - 3k)

= (1+0)i + (3-1)j + (5-3)k

A + B = i + 2j + 2k

Magnitude of Vector R = |R|

|\vec {R}| = \sqrt{a^2 + b^2 + c^2} \\=  \sqrt{1^2 + 2^2 + 2^2}\\=\sqrt{9}

= 3

\hat{R} = \vec{R} / |R|

\hat{R}=\frac{3i + 4j + 5k}{3}

          = (1/3) i + (2/3) j + (2/3)k

Practice Problems on Unit Vector

1. Given the vector v = 3\hat{i} + 4\hat{j} ​, find the unit vector in the direction of v.

2. Given the vector u = \hat i - 2\hat j + 2\hat k, find the unit vector in the direction of u.

3. Determine whether the vector w = \frac 1{\sqrt 2} \hat i + \frac 1{\sqrt 2} \hat j ​is a unit vector.

4. Find the unit vector in the direction of the vector a = -7\hat i + 24 \hat j.