# Write all the unit vectors in XY- plane.

Answer

Verified

279k+ views

**Hint:**For solving this question you should know about the unit vectors in planes. A unit vector in the XY- plane indicates that there is no Z-axis, we have to find the value of $\overrightarrow{a}=x\widehat{i}+y\widehat{j}$ and for that we will let a unit vector $\overrightarrow{a}$ and using the formula of dot product $\overrightarrow{a}.\overrightarrow{b}=\left| \overrightarrow{a} \right|\left| \overrightarrow{b} \right|\cos \theta $ we will find the value of X and Y.

**Complete step-by-step solution:**

According to our question it is asked to write all the unit vectors in the XY- plane. Let the unit vector be $\overrightarrow{a}$, we know that $\overrightarrow{a}=x\widehat{i}+y\widehat{j}+z\widehat{k}$, where x-axis is $\widehat{i}$, y-axis is $\widehat{j}$ and z-axis is$\widehat{k}$. Since the vector is in XY-plane, that means there is no z-coordinate (z = 0). Then,

$\begin{align}

& \overrightarrow{a}=x\widehat{i}+y\widehat{j}+0\widehat{k} \\

& \Rightarrow \overrightarrow{a}=x\widehat{i}+y\widehat{j} \\

\end{align}$

The unit vector in the direction of x-axis is $\widehat{i}$ and y-axis is $\widehat{j}$.

Angle with x-axis:

Since $\overrightarrow{a}$ makes an angle of $\theta $ with x-axis, so angle between $\overrightarrow{a}$ and $\widehat{i}$ is $\theta $.

We know that: $\overrightarrow{a}.\overrightarrow{b}=\left| \overrightarrow{a} \right|\left| \overrightarrow{b} \right|\cos \theta $

So, putting $\overrightarrow{a}=\overrightarrow{a,}\overrightarrow{b}=\widehat{i,}\theta =\theta $

$\overrightarrow{a}.\widehat{i}=\left| \overrightarrow{a} \right|\left| \widehat{i} \right|\cos \theta $

As $\overrightarrow{a}$ is a unit vector, $\left| \overrightarrow{a} \right|=1$ and $\widehat{i}$ is a unit vector, $\left| \widehat{i} \right|=1$. So,

$\begin{align}

& \overrightarrow{a}.\widehat{i}=1\times 1\times \cos \theta \\

&\Rightarrow \overrightarrow{a}.\widehat{i}=\cos \theta \\

&\Rightarrow \left( x\widehat{i}+y\widehat{j}+0\widehat{k} \right).\widehat{i}=\cos \theta \\

&\Rightarrow \left( x\widehat{i}+y\widehat{j}+0\widehat{k} \right).\left( 1\widehat{i}+0\widehat{j}+0\widehat{k} \right)=\cos \theta \\

&\Rightarrow x.1+y.0+0.0=\cos \theta \\

&\Rightarrow x=\cos \theta \\

\end{align}$

Angle with y-axis:

Since $\overrightarrow{a}$ makes an angle of $\left( {{90}^{\circ }}-\theta \right)$ with y-axis, so angle between $\overrightarrow{a}$ and $\widehat{j}$ is $\left( {{90}^{\circ }}-\theta \right)$.

We know that: $\overrightarrow{a}.\overrightarrow{b}=\left| \overrightarrow{a} \right|\left| \overrightarrow{b} \right|\cos \theta $

So, putting $\overrightarrow{a}=\overrightarrow{a,}\overrightarrow{b}=\widehat{j,}\theta =\left( {{90}^{\circ }}-\theta \right)$

$\begin{align}

& \overrightarrow{a}.\widehat{j}=\left| \overrightarrow{a} \right|\left| \widehat{j} \right|\cos \left( {{90}^{\circ }}-\theta \right) \\

&\Rightarrow \overrightarrow{a}.\widehat{j}=1\times 1\times \cos \left( {{90}^{\circ }}-\theta \right) \\

&\Rightarrow \overrightarrow{a}.\widehat{j}=\cos \left( {{90}^{\circ }}-\theta \right) \\

&\Rightarrow \overrightarrow{a}.\widehat{j}=\sin \theta \\

&\Rightarrow \left( x\widehat{i}+y\widehat{j}+0\widehat{k} \right).\widehat{j}=\sin \theta \\

&\Rightarrow \left( x\widehat{i}+y\widehat{j}+0\widehat{k} \right).\left( 0\widehat{i}+1\widehat{j}+0\widehat{k} \right)=\sin \theta \\

&\Rightarrow x.0+y.1+0.0=\sin \theta \\

&\Rightarrow y=\sin \theta \\

\end{align}$

Thus,

$\begin{align}

& \overrightarrow{a}=x\widehat{i}+y\widehat{j} \\

&\Rightarrow \overrightarrow{a}=\cos \theta \widehat{i}+\sin \theta \widehat{j} \\

\end{align}$

This value will be true in all quadrants. So, $0\le \theta \le 2\pi $.

**Therefore $\overrightarrow{a}=\cos \theta \widehat{i}+\sin \theta \widehat{j}$; for $0\le \theta \le 2\pi $.**

**Note:**While solving this type of questions you should be careful of the angles from the x-axis and y-axis and always, we have to make a unit vector. And this will be mandatory to be in the XY-plane. And if it is in the XY-plane, then the rest Z-plane coordinate must be always zero.

Recently Updated Pages

Basicity of sulphurous acid and sulphuric acid are

Why should electric field lines never cross each other class 12 physics CBSE

An electrostatic field line is a continuous curve That class 12 physics CBSE

What are the measures one has to take to prevent contracting class 12 biology CBSE

Suggest some methods to assist infertile couples to class 12 biology CBSE

Amniocentesis for sex determination is banned in our class 12 biology CBSE

Trending doubts

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

How many meters are there in a kilometer And how many class 8 maths CBSE

What is pollution? How many types of pollution? Define it

Change the following sentences into negative and interrogative class 10 english CBSE

What were the major teachings of Baba Guru Nanak class 7 social science CBSE

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Draw a labelled sketch of the human eye class 12 physics CBSE