Unit Vector

Unit vectors are the vectors that have a magnitude (length) of exactly 1.

• It shows direction only, not size.
• Any vector can be converted into a unit vector by dividing it by its magnitude.
• Represented by the symbol '^' such as \hat{a}.

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 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 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

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.

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:

\hat{v} = \frac{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:

\hat{w} = \frac{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.

Also Check

• 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.