To understand a unit vector, you should know what a vector is? In simple words, a vector is an entity that depends on the magnitude as well as direction, for example, displacement, work, etc. If any of the quantities (direction or magnitude) changes, the value of the vector also changes. On the other hand, scalar quantity only depends on the magnitude, which means that direction doesn't matter in scalar quantity like time, mass, speed, etc. but this topic needs a separate blog for discussion, let's just stick with the vectors for now.

What is a Unit Vector?

Vectors come in all kinds of shapes and sizes but how do we describe those vectors? With the help of the unit vector. The definition of the unit vector is pretty simple, it is a vector that has a magnitude of 1. This means that any vector which has a magnitude of one, irrespective of direction, is called a unit vector. Let's say we have a unit vector and we named it vector A. Vector A is a unit vector but for someone who is trying to understand your work will have a difficult time and that is why a unit vector has a representation. We represent unit vectors like this:

\hat { A }

The cap you are seeing above the letter "A" is called a hat. If you see a hat on any vector, it means that the vector is a unit vector. Unit vectors have many uses, let's talk about that.

Representation of Vectors

For example, you have a vector \overrightarrow { A } =\begin{pmatrix} 2 \\ -3 \end{pmatrix}, this means the head of the vector is 2 unit long in the positive direction and the tail of the vector is 3 unit long in the negative direction. The question is, how will you express it in unit vector? For this, we need to draw a frame of reference. Let's say is the unit vector along the x-axis and is the unit vector along the y-axis. Since the vector \overrightarrow { A } has positive 2 units in the x-axis direction, it is equivalent to 2i. The vector \overrightarrow { A } also has 3 units in the y-axis but in the negative direction, it will be equivalent to the 3but our work here is not done, remember the negative direction? We need to add that too when we are representing any vector into its unit vector. Hence, the y-axis of the vector \overrightarrow { A } will be equal to -3j.

Last but not least, its time to write the \overrightarrow { A } into its unit vector. This is how you will represent vector \overrightarrow { A } into its unit vectors:

\overrightarrow { A } =\begin{pmatrix} 2 \\ -3 \end{pmatrix} = 2i - 3j

Although, the above vector was in 2 dimensions, however, vectors also exist in 3 dimensions as well. The third dimension is called the z-axis and there is nothing to worry about, it will also be written in the same way. Below are some examples of 2d and 3d vector representation into its unit vectors.

 

1. \overrightarrow { Y } =\begin{pmatrix} 4 \\ 5 \end{pmatrix}

i = 4

j = 5

k = 0

\overrightarrow { Y } =\begin{pmatrix} 4 \\ 5 \end{pmatrix} = 4i + 5j

 

2. \overrightarrow { B } =\begin{pmatrix} -2 \\ -1 \end{pmatrix}

i = -2

j = -1

k = 0

\overrightarrow { B } =\begin{pmatrix} -2 \\ -1 \end{pmatrix} = -2i - 1j

 

3. \overrightarrow { J } =\begin{pmatrix} -4 \\ 4 \\ -3 \end{pmatrix}

i = -4

j = 4

k = -3

\overrightarrow { J } =\begin{pmatrix} -4 \\ 4 \\ -3 \end{pmatrix} = -4i + 4j - 3k

 

4. \overrightarrow { H } =\begin{pmatrix} 0 \\ 0 \\ -1 \end{pmatrix}

i = 0

j = 0

k = -1

\overrightarrow { H } =\begin{pmatrix} 0 \\ 0 \\ -1 \end{pmatrix} = -k

Scaling

Unit vectors can also be scaled to get a new vector. Basically, the unit vector will be multiplied by a scalar quantity which will increase the magnitude of the vector, however, the direction will remain unaffected. That is the beauty of the scaling vectors. For example, you have a unit vector \hat { a }, if we multiply it with a scalar quantity, let's say 2, it will result in a new vector.

\overrightarrow { a } = 2 \times \hat { a }

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Normalizing a Vector

Till here, you have learned how a unit vector is used in different ways but some of you might be thinking about how to get a unit vector from a vector? The answer is by normalizing a vector. This method is purely developed to obtain the unit vector of any vector. The concept of this method is to change the magnitude of any vector. The direction of the vector will remain the same but when you normalize the vector, it will change the magnitude of the specific vector. To do that, every element of the vector will be divided by its magnitude. Here is how to do it:

\hat { A } = \frac { \overrightarrow { A }  }{ \left| \overrightarrow { A } \right| }

To find the magnitude of any vector, you need to square all the elements of the specific vector and then add them. Take the square root of the answer and whatever result you will get is the magnitude of the vector. In simple words, you need to use the Pythagorean theorem to find the magnitude of the vector.

c = \sqrt { { a }^{ 2 } + { b }^{ 2 } }

In the case of vector: \left| \overrightarrow { A } \right| = \sqrt { ({ A }_{ x } \times { A }_{ x }) + ({ A }_{ y } \times { A }_{ y }) }

Examples

If \overrightarrow { v } is a vector of components (3, 4), find a unit vector in the same direction.

\overrightarrow { v } = (3, 4)

 

\left| \overrightarrow { v } \right| = \sqrt { { 3 }^{ 2 } + { 4 }^{ 2 } } = 5

\hat { v } = \frac { 1 }{ 5 } \times (3, 4)

\hat { v } = (\frac { 3 }{ 5 }, \frac { 4 }{ 5 } )

 

Find the unit vector \overrightarrow { u } with the direction of vector \overrightarrow { u } = 8i - 6j

\left| \overrightarrow { v } \right| = \sqrt { { 8 }^{ 2 } + { 6 }^{ 2 } } = 10

 

\hat { u } = \frac { 8i - 6j }{ 10 }

\hat { u } = \frac { 4 }{ 5 } i - \frac { 3 }{ 5 } j

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.