Answer
Verified
417.6k+ views
Hint: If a unit vector is making an angle $\theta $ with the positive direction of x-axis, then it’s component along the x-axis is $\cos \theta $ and it’s component along the positive y-axis is $\sin\theta $. This unit vector will be written as $\cos \theta \widehat{i}+\sin \theta \widehat{j}$.
Before proceeding with the question, we must know the formula that will be required to solve this question. In vectors, if a unit vector is making an angle $\theta $ with the positive x-axis, then the vector can be written as $\cos \theta \widehat{i}+\sin \theta \widehat{j}$. The x component of this vector is $\cos \theta $ and the y component of this vector is $\sin \theta $.
In the question, we have to write down the unit vector making an angle ${{30}^{\circ }}$ with the positive x-axis.
The x component of this vector will be $\cos {{30}^{\circ }}$ and the y component of this vector will be $\sin {{30}^{\circ }}$. So, this unit vector which is making an angle ${{30}^{\circ }}$ with the positive x-axis can be written as \[\cos {{30}^{\circ }}\widehat{i}+\sin {{30}^{\circ }}\widehat{j}\].
From trigonometry, we have $\cos {{30}^{\circ }}=\dfrac{\sqrt{3}}{2}$ and $\sin {{30}^{\circ
}}=\dfrac{1}{2}$. Substituting these values in the above vector, we get,
$\dfrac{\sqrt{3}}{2}\widehat{i}+\dfrac{1}{2}\widehat{j}$
There is one more possible vector that can make an angle of ${{30}^{\circ }}$ with the positive x-axis.
That vector is as shown below,
The x component of this vector will be $\cos {{30}^{\circ }}$ and the y component of this vector will be $\sin {{30}^{\circ }}$. But in this vector, the y component will be along the negative y-axis. So, this unit vector which is making an angle ${{30}^{\circ }}$ with the positive x-axis can be written as
\[\cos {{30}^{\circ }}\widehat{i}-\sin {{30}^{\circ }}\widehat{j}\].
From trigonometry, we have $\cos {{30}^{\circ }}=\dfrac{\sqrt{3}}{2}$ and $\sin {{30}^{\circ
}}=\dfrac{1}{2}$. Substituting these values in the above vector, we get,
$\dfrac{\sqrt{3}}{2}\widehat{i}-\dfrac{1}{2}\widehat{j}$
So, the two possible answers are $\dfrac{\sqrt{3}}{2}\widehat{i}+\dfrac{1}{2}\widehat{j}$ and $\dfrac{\sqrt{3}}{2}\widehat{i}-\dfrac{1}{2}\widehat{j}$.
Note: In this question, we were given a unit vector which is making an angle of ${{30}^{\circ }}$ with the positive x-axis. We wrote this vector by using the formula $\cos \theta \widehat{i}+\sin \theta \widehat{j}$. But in case we are given a vector which is also having a magnitude, let us say $r$and making an angle $\theta $ with the positive x-axis, we will write that vector by using the formula
$r\left( \cos \theta \widehat{i}+\sin \theta \widehat{j} \right)$.
Before proceeding with the question, we must know the formula that will be required to solve this question. In vectors, if a unit vector is making an angle $\theta $ with the positive x-axis, then the vector can be written as $\cos \theta \widehat{i}+\sin \theta \widehat{j}$. The x component of this vector is $\cos \theta $ and the y component of this vector is $\sin \theta $.
In the question, we have to write down the unit vector making an angle ${{30}^{\circ }}$ with the positive x-axis.
The x component of this vector will be $\cos {{30}^{\circ }}$ and the y component of this vector will be $\sin {{30}^{\circ }}$. So, this unit vector which is making an angle ${{30}^{\circ }}$ with the positive x-axis can be written as \[\cos {{30}^{\circ }}\widehat{i}+\sin {{30}^{\circ }}\widehat{j}\].
From trigonometry, we have $\cos {{30}^{\circ }}=\dfrac{\sqrt{3}}{2}$ and $\sin {{30}^{\circ
}}=\dfrac{1}{2}$. Substituting these values in the above vector, we get,
$\dfrac{\sqrt{3}}{2}\widehat{i}+\dfrac{1}{2}\widehat{j}$
There is one more possible vector that can make an angle of ${{30}^{\circ }}$ with the positive x-axis.
That vector is as shown below,
The x component of this vector will be $\cos {{30}^{\circ }}$ and the y component of this vector will be $\sin {{30}^{\circ }}$. But in this vector, the y component will be along the negative y-axis. So, this unit vector which is making an angle ${{30}^{\circ }}$ with the positive x-axis can be written as
\[\cos {{30}^{\circ }}\widehat{i}-\sin {{30}^{\circ }}\widehat{j}\].
From trigonometry, we have $\cos {{30}^{\circ }}=\dfrac{\sqrt{3}}{2}$ and $\sin {{30}^{\circ
}}=\dfrac{1}{2}$. Substituting these values in the above vector, we get,
$\dfrac{\sqrt{3}}{2}\widehat{i}-\dfrac{1}{2}\widehat{j}$
So, the two possible answers are $\dfrac{\sqrt{3}}{2}\widehat{i}+\dfrac{1}{2}\widehat{j}$ and $\dfrac{\sqrt{3}}{2}\widehat{i}-\dfrac{1}{2}\widehat{j}$.
Note: In this question, we were given a unit vector which is making an angle of ${{30}^{\circ }}$ with the positive x-axis. We wrote this vector by using the formula $\cos \theta \widehat{i}+\sin \theta \widehat{j}$. But in case we are given a vector which is also having a magnitude, let us say $r$and making an angle $\theta $ with the positive x-axis, we will write that vector by using the formula
$r\left( \cos \theta \widehat{i}+\sin \theta \widehat{j} \right)$.
Recently Updated Pages
Basicity of sulphurous acid and sulphuric acid are
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
What is the stopping potential when the metal with class 12 physics JEE_Main
The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main
Using the following information to help you answer class 12 chemistry CBSE
Trending doubts
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE
Select the correct plural noun from the given singular class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
The sum of three consecutive multiples of 11 is 363 class 7 maths CBSE
What is the z value for a 90 95 and 99 percent confidence class 11 maths CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
How many squares are there in a chess board A 1296 class 11 maths CBSE