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Unit Vectors

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What is Unit Vector in Physics?

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A vector quantity bears both magnitude and direction. For example, displacement is the shortest distance taken to reach your destination. It is displayed as a displacement vector. 

A vector also carries a negative magnitude when an object takes the opposite direction. There is one more term called a unit vector. Unit vectors have a magnitude of 1.

A speciality of a unit vector in Physics is, a vector can be represented in the space using a unit vector. 

On this page, we will understand what is unit vector in Physics Class 11, and the unit vector definition in Physics in detail.


Define Unit Vector Class 11

A unit vector definition in Physics is, a unit vector is a direction vector. A unit vector symbol is similar to the mathematical symbol of the exponent, and that is ‘^’. Unit vector symbol in Physics is pronounced as ‘cap or hat’. 

For calculating the magnitude of any given vector, we use the coordinate system as follows:

[Image will be Uploaded Soon] 

Here, 

A unit vector, i cap indicates the direction of an object along the x-axis.  

A unit vector, j cap indicates the direction of an object along the y-axis. 

A unit vector, k cap indicates the direction of an object along the z-axis.

So, do you know what is unit vector in Physics?

Well! We define unit vector in Physics with the following equation:

Unit vector = \[\frac{vector}{\text{Magnitude of the vector}}\] 

So, a unit vector \[\hat{p}\] having the same direction as a vector \[\vec{p}\] is given as;

\[\hat{p}\] = \[\vec{p}\] |\[\vec{p}\]|

Here,

\[\hat{p}\] =  a unit vector

\[\vec{p}\] =  represents the vector

|\[\vec{p}\]| =  represents the magnitude of the vector


Components of a Unit Vector

In the Cartesian coordinate system, any vector \[\vec{q}\] can be represented in terms of its unit vectors. The unit vectors in direction of x, y, and z-axes are stated as;

i cap, j cap, and k cap, respectively. The unit vector \[\vec{q}\] can be represented in space with respect to the origin of the given coordinate system as;

\[\vec{q}\] = x\[\hat{i}\]  + y\[\hat{j}\] + z\[\hat{k}\]

The unit vector \[\vec{q}\] can be resolved into components along the three axes as shown in the given figure. With OM as the diagonal, a parallel pipe is constructed whose edges are: OA, OB, and OC that lie along the three perpendicular axes.

\[\vec{OA}\] = x\[\hat{i}\] 

\[\vec{OB}\] = y\[\hat{j}\]

\[\vec{OC}\] = z\[\hat{k}\]

The vector \[\vec{q}\] can be represented along OM is given as;

\[\vec{OM}\] = x\[\hat{i}\] + y\[\hat{j}\] + z\[\hat{k}\]


Point to Note:

We must ensure that any two unit vectors p̂ and q̂ must not be considered equal just because they have equal magnitudes. Since the direction in which the vectors are taken might be varying; therefore these unit vectors are different from each other. Therefore, unit vector definition in physics should specify both magnitude and direction.


Unit Vector in Physics 

A unit vector in Physics Class 11 can be determined in the following manner:

For fiding a unit vector ‘p’,  in the same direction of a vector ‘q,’ we divide the vector by its magnitude as;

\[\vec{p}\] = \[\frac{\vec{q}}{\mid \mid \vec{q} \mid \mid }\] 

We can rewrite the above equation as;

\[\vec{p}\] = \[\frac{1}{\mid \mid \vec{q} \mid \mid }\] ∗ (\[\vec{q}\])

For a vector \[\vec{q}\] = <a, b>, the magnitude is given by:

|| \[\vec{q}\] || = \[\sqrt{a^{2} + b^{2}}\]

So, finding the magnitude of a given vector with its components, we can use the Pythagoras theorem, as we used above.

For example, if a unit vector \[\vec{r}\] in the same direction as B = (8, 6), what will be the value of the unit vector \[\vec{r}\]?

Now, let us understand it by applying the Pythagoras theorem:

Here, we have <a, b> = <8, 6>

And,

<a, b> = \[\sqrt{a^{2} + b^{2}}\]

a = 8, b = 8

Putting the values of ‘a’ and ‘b,’ in the above equation, we have;

= \[\sqrt{8^{2} + 6^{2}}\] = \[\sqrt{100}\] = 10

So, the value of the unit vector || \[\vec{r}\] || = 10.


Unit Vectors in Mathematical Terms

In mathematics, unit vectors are referred to as the normal vector space a.k.a a spatial vector of length 1. Furthermore, we use a lowercase letter with a circumflex, or ‘hat’, pronounced as “i-hat”). 

We use the term direction vector to characterize a unit vector to represent the spatial vector. Besides this, we denote them as ‘d’. 

Also, 2-dimensional spatial directions represent the numerical points on the unit circle in the following way:

[Image will be Uploaded Soon] 

FAQ (Frequently Asked Questions)

Q1: What is a Unit Vector and Why Do We Use it for?

Ans: As you know, unit vectors can be compared to units of physical quantities.

When we say 6 m, what does that exactly mean?

Here, ‘m’ denotes the measure of length.

Also, we can say this tells us that it's 6 times the unit length of magnitude 1, I.e., 6 * 1m

A similar case is with vectors.

When we say 6î, what does that exactly mean?

Here, î denotes the direction of the vector component.

Also, we can say this tells us that it's 6 times the unit vector of magnitude 1, I.e., 6 * 1î.

So just like we use m to denote that what kind of quantity it is, the unit vector is used to specify the direction of the vector component.

As we know that ‘m’ is a fixed unit of the quantity of unit magnitude, similarly,  the meaning of unit vectors is a fixed vector of unit magnitude.


Q2: Specify if the Area is a Scalar or a Vector Quantity.

Ans: An instance in physics where the flux is used is Gauss’s Law (in the integral form), which states that the net electric flux via a closed surface is proportional to the charge enclosed by it:

                       ∮E⋅dA = qen0

In general, the area is a scalar but in three dimensions we treat it as a vector for the purpose of taking surface integrals which then follows a process similar to that of taking line integrals. 

Therefore, treating the area as a vector field is necessary because the total area does not give the full description of a surface. So, despite having the same area, we get typically different results to given physics problems. Since area usually refers to the total area of a surface, However, scalar would be the most appropriate term for it.