Unit Vector

Dhristi JEE 2022-24

What is Unit Vector?

Vector units have both direction and magnitude. However, sometimes one is interested only in direction and not the magnitude. In such a case, vectors are often considered unit length. These unit vectors are generally used to represent direction, with a scalar coefficient providing the magnitude. A vector decomposition can be expressed as a sum of unit vector and scalar coefficients.

Vector units are often used to represent quantities in Physics such as force, acceleration, quantity, or torque. In this article, we will discuss how to find unit vectors.

The geometrics entities that have both magnitude and direction are known as vectors. Vectors start from a starting point and reach the terminal point, which represents the final position. Vectors can be added, subtracted or multiplicated. 

The vector having a magnitude of 1 is known as the unit vector. A vector which when divided by the magnitude of the same given vector gives a unit vector. Unit vectors are also known as direction vectors. Unit vectors are denoted by \[\hat{a}\] and their lengths are equal to 1.

Magnitude of Unit Vector

In order to calculate the numeric value of a given vector, the magnitude of the vector formula is used. The magnitude of a vector \[\vec{A}\]is |A|. The magnitude of a vector can be identified by calculating the square roots of the sum of squares of its direction vectors. The magnitude of a vector formula is given by:

|A| = \[\sqrt{a_{1}^{2} + b_{1}^{2} + c_{1}^{2}}\]

The unit vector is denoted by ‘^’, which is called a hat or cap. For example, if the unit vector is \[\hat{A}\], it will be read as A cap.

For a unit vector u in the same direction as vector v, we divide the vector by its magnitude

\[\vec{u} = \frac{\vec{v}}{|| \vec{v}||} = \frac{1}{|| \vec{v}||} \vec{v}\]

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

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

\[\text{Unit vector} = \frac{\text{Vector}}{\text{Vector's magnitude}}\]

Representation of a Vector

There are two ways in which a vector can be represented-

  1. \[\vec{a}\] = (x,y,z)

  2. \[\vec{a} = \widehat{xi} + \widehat{yj} + \widehat{zk}\]

Unit Normal Vector

The vector which is perpendicular to the surface at a defined point  is defined as the ‘normal’ vector. It can also be stated as a vector normal to the surface which contains the vector. After normalizing the normal vector, the unit vector which is acquired is known as the unit normal vector, sometimes called unit normal.

Solved Examples Related to Unit Vector

Example 1.  Find the unit vector in the direction of vector \[\vec{a} = \widehat{2i} +  \widehat{3j} + \hat{k}.\]

Solution: Given,

\[\vec{a} = \widehat{2i} +  \widehat{3j} + \hat{k}.\]

Magnitude of \[\vec{a} = \sqrt{2^{2} + 3^{2} + 1^{2}}\]

\[|\vec{a}|= \sqrt{4+9+1}\]

\[|\vec{a}|= \sqrt{14}\]

Unit vector in direction \[\vec{a} = \frac{1}{Magnitude of \vec{a}} \times \vec{a}\]

\[\hat{a} = \frac{1}{\sqrt{14}}[\widehat{2i}+\widehat{3j}+\hat{k}]\]

\[\hat{a} = \frac{2}{\sqrt{14}}\hat{i} + \frac{3}{\sqrt{14}}\hat{j} + \frac{1}{\sqrt{14}}\hat{k}\]

Example 2. Determine the unit vector in the direction of sum of vectors, \[\vec{a} = \widehat{2i} + \widehat{2j} - \widehat{5k}\] & \[\vec{b} = \widehat{2i} + \widehat{j} + \widehat{3k}\].

Solution: Given 

\[\vec{a} = \widehat{2i} + \widehat{2j} - \widehat{5k}\]

\[\vec{b} = \widehat{2i} + \widehat{j} + \widehat{3k}\]

Let \[\vec{c} = (\vec{a} + \vec{b})\]

\[\vec{c} = (2 + 2) \hat{i} + (2+1) \hat{j} + (-5 + 3)\hat{k}\]

\[\vec{c} = \widehat{4i} + \widehat{3j} - \widehat{2k}\]

Magnitude of \[\vec{c} = \sqrt{4^{2} + 3^{2} + (-2)^{2}}\]

\[|\vec{c}| = \sqrt{16 + 9 + 4}\]

\[|\vec{c}| = \sqrt{29}\]

Unit vector in direction of \[\vec{c} = \frac{1}{|\vec{c}|}\vec{c}\]

\[\hat{c} = \frac{1}{\sqrt{29}} [\widehat{4i} +\widehat{3j} - \widehat{2k}]\]   

\[\hat{c} = \frac{4}{\sqrt{29}}\hat{i} + \frac{3}{\sqrt{29}}\hat{j} - \frac{2}{\sqrt{29}}\hat{k}\]

Spherical Coordinate Unit Vector

The unit vectors in spherical coordinate systems are defined as the function of position. It is convenient to express spherical coordinate unit vectors in terms of rectangular coordinate systems which are not themself the function of position.

\[\hat{r}\] = \[\frac{\hat{r}}{r}\] = \[\frac{x\hat{x} + y\hat{y} + z\hat{z}}{r}\] = \[\hat{x}\] sinθ cosØ + \[\hat{y}\]sinθ sinØ - \[\hat{z}\]cosθ

\[\frac{\hat{z \times r}}{Sin \theta}\] = - \[\hat{x}\] sinØ + \[\hat{y}\] sinØ 

\[\hat{\theta}\] = \[\hat{\varnothing }\] x \[\hat{r}\] = \[\hat{x}\]cos θ cos Ø + \[\hat{y}\]cos θ sin Ø - \[\hat{z}\] sin θ

Unit Tangent Vector

Considering a smooth vector-valued function \[\vec{V}\](t), any vector parallel to \[\vec{V}\]’(t₀) is considered as the tangent to the graph of \[\vec{V}\](t) at t = t₀. It is generally used to represent direction \[\vec{V}\]’(t), not magnitude. 

Hence, we are considering the unit vector in the direction of \[\vec{V}\]’(t). This gives the unit tangent vector definition as follows:

Let us consider \[\vec{V}\]’(t) as a smooth function on the open interval I. Then, the unit tangent vector \[\hat{t}\](t) in this case is:

 \[\hat{t}\](t) = \[\vec{V}\]’(t) / |\[\vec{V}\]’(t)|

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FAQs (Frequently Asked Questions)

1. What is the dot product?

The dot product or scalar product of two non-zero vectors\[\vec{a}\] and \[\vec{b}\] , denoted by

\[\vec{a}.\vec{b}\]  is 

\[\vec{a}.\vec{b} = |\vec{a}||\vec{b}| cos \theta\]

where θ is the angle in between of  \[\vec{a}\] and \[\vec{b}\] and 0 \[\leq \theta \leq \pi\]

If \[\vec{a} = 0\] or \[\vec{b} = 0\], then θ is not defined and then

\[\vec{a}.\vec{b}\] =0

Properties of Dot Product:

  • When \[\vec{a}, \vec{b}, and \vec{c}\] are any three vectors, then 

\[\vec{a}.(\vec{b}+\vec{c}) = \vec{a}.\vec{b} + \vec{a}.\vec{c}\]

  • When \[\vec{a}\] and \[\vec{b}\]are any two vectors and is a scalar, then

\[(\lambda \vec{a}). \vec{b} = \lambda(\vec{a}.\vec{b}) = \vec{a}.(\lambda \vec{b}\]

2. What is a cross product?

Cross product, also known as the vector product, is the product of vectors having different kinds of nature. When two vectors are multiplied and the product also gives a vector quantity, then the resultant vector is said to be the cross product of the two vectors.

The resultant vector in a cross product is perpendicular to the plane which contains the two given vectors.

When two independent vectors \[\vec{A}\] and \[\vec{B}\] are multiplied then the result of cross product of the vectors \[\vec{A} \times \vec{B}\],  is perpendicular to both the vectors and the plane containing the two given vectors. 

The cross product is represented by:

\[\vec{A} \times \vec{B} = |\vec{A}| \times |\vec{B}|\] sin θ

3. What are the properties of a unit vector?

The properties of a unit vector are-

  • The magnitude of a unit vector is always 1.

  • The directions of vectors can be specified with the help of unit vectors.

  • Unit vectors exist in both 2-D and 3-D.

  • Unit vectors are present in every vector in the form of its component.

  • In a vector, the unit vector is directed along its axes.

  • In a vector, the unit vectors are normal to corresponding unit vectors of the same vector.

4. What are the Algebraic properties of a vector?

Some algebraic properties of a vector are-

  • Commutative vector: P+Q=Q+P

  • Associative vector: (P+Q)+R=P+(R+Q)

  • Additive Identity: A vector 0 is there such that (P+0)=P=(0+P) for all P

  • Additive Inverse: For any P, -P vector is there such that P+(-P)=0

  • Distributive Vector: r(P+Q)=rP+qP

  • Distributive (scalar): (r+s)P=rP+sP

  • Associative (scalar): r(sP)=(rs)P

  • Multiplicative identity for 1 which is a real number: 1P=P for each P

5. What are some applications of vectors in real life?

Situations in real life that involve force or velocity need the applications of vectors. Some of them are-

  • Forces acting on a boat while crossing the river.

  • Vectors are used in sports such as a quarterback’s pass which has direction as well as magnitude.

  • Wind vectors tell the number of different geographic locations.

  • The billiards also uses the concept of vector since it has direction and magnitude both.

6. What is a Unit Vector in Physics?

A unit vector also known as a directional vector is a vector with a magnitude of 1 unit. It is denoted using a lowercase letter with a cap symbol (‘^’) along with it. Any vector can be easily converted into a unit vector by dividing it by vector magnitude as given below.

Unit Vector = Vector / Vector’s Magnitude

7. Does the Unit Vector Have Any Units?

No, the unit vector has only directions but no unit or dimensions.

8. Can Unit Vectors be Negative?

Two vectors are considered equivalent if they represent the same direction and magnitude. Similar to the scalar which can be either positive or negative, the vectors can also be negative or positive.