If you happen to be viewing the article What is Unit Vector?? on the website Math Hello Kitty, there are a couple of convenient ways for you to navigate through the content. You have the option to simply scroll down and leisurely read each section at your own pace. Alternatively, if you’re in a rush or looking for specific information, you can swiftly click on the table of contents provided. This will instantly direct you to the exact section that contains the information you need most urgently.
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.
Contents
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-
-
\[\vec{a}\] = (x,y,z)
-
\[\vec{a} = \widehat{xi} + \widehat{yj} + \widehat{zk}\]
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 \[\vec{V}\]
Thank you so much for taking the time to read the article titled What is Unit Vector? written by Math Hello Kitty. Your support means a lot to us! We are glad that you found this article useful. If you have any feedback or thoughts, we would love to hear from you. Don’t forget to leave a comment and review on our website to help introduce it to others. Once again, we sincerely appreciate your support and thank you for being a valued reader!
Source: Math Hello Kitty
Categories: Math
