Answer
398.1k+ views
Hint: In this question we are asked to find the component form of the vector which is given in form $ v = \left| v \right|\cos \theta i + \left| v \right|\sin \theta j $ , where $ \left| v \right| $ is the magnitude and $ \theta $ is the angle made with the $ x $ -axis, now substituting the values given in the component form, and using the values of trigonometric ratios of cos and sin we will get the required component form.
Complete step by step answer:
The component form of the vector $ v $ is given by $ v = \left| v \right|\cos \theta i + \left| v \right|\sin \theta j $ , where $ \left| v \right| $ is the where $ \left| v \right| $ is the magnitude and\[\theta \]is the angle made with the $ x $ -axis.
Now in the given question, the data given is, $ \theta = {45^o} $ , and the magnitude i..e, $ \left| v \right| $ is equal to 1.
Now substituting the given values in the component form we get,
$ v = \left| v \right|\cos \theta i + \left| v \right|\sin \theta j $ ,
By substituting the values we get,
$ \Rightarrow v = \left( 1 \right)\cos \left( {{{45}^0}} \right)i + \left( 1 \right)\sin \left( {{{45}^0}} \right)j $ ,
Now from the trigonometric ratios we know that, $ \cos {45^0} = \dfrac{1}{{\sqrt 2 }} $ and $ \sin {45^0} = \dfrac{1}{{\sqrt 2 }} $ ,
Now substituting the cos and sin values we get,
\[ \Rightarrow v = \left( 1 \right)\dfrac{1}{{\sqrt 2 }}i + \left( 1 \right)\dfrac{1}{{\sqrt 2 }}j\],
Now simplifying we get,
$ \Rightarrow v = \dfrac{1}{{\sqrt 2 }}i + \dfrac{1}{{\sqrt 2 }}j $ ,
So, the component form is equal to $ v = \dfrac{1}{{\sqrt 2 }}i + \dfrac{1}{{\sqrt 2 }}j $ .
This can be represented using the graph,
The component form can also be written in the form $ \left( {\dfrac{1}{{\sqrt 2 }},\dfrac{1}{{\sqrt 2 }}} \right) $ .
$ \therefore $ The component form of the vector magnitude 1 and the angle it makes with the positive $ x $ -axis is $ \theta = {45^o} $ is equal to $ v = \dfrac{1}{{\sqrt 2 }}i + \dfrac{1}{{\sqrt 2 }}j $ , the component form can also be written as,\[\left( {\dfrac{1}{{\sqrt 2 }},\dfrac{1}{{\sqrt 2 }}} \right)\].
Note: For resolving a vector into its components, we can use the following formulas:
Consider $ a $ to be the magnitude of the vector $ \overrightarrow a $ and $ \theta $ to be the angle that is formed by the vector along the $ x $ -axis or to be the direction of the given vector. Then we will get,
\[{a_x} = \left| a \right| \times \cos \theta \]and $ {a_y} = \left| a \right| \times \sin \theta $ ,
Where $ {a_x} $ is called the magnitude of the $ x $ -component of the given vector $ \overrightarrow a $ ,
Where $ {a_y} $ is called the magnitude of the $ y $ -component of the given vector $ \overrightarrow a $ .
And the vector form is written as, $ \overrightarrow a = \left( {{a_x},{a_y}} \right) = \left| a \right|\cos \theta i + \left| a \right|\sin \theta j $ .
Complete step by step answer:
The component form of the vector $ v $ is given by $ v = \left| v \right|\cos \theta i + \left| v \right|\sin \theta j $ , where $ \left| v \right| $ is the where $ \left| v \right| $ is the magnitude and\[\theta \]is the angle made with the $ x $ -axis.
Now in the given question, the data given is, $ \theta = {45^o} $ , and the magnitude i..e, $ \left| v \right| $ is equal to 1.
Now substituting the given values in the component form we get,
$ v = \left| v \right|\cos \theta i + \left| v \right|\sin \theta j $ ,
By substituting the values we get,
$ \Rightarrow v = \left( 1 \right)\cos \left( {{{45}^0}} \right)i + \left( 1 \right)\sin \left( {{{45}^0}} \right)j $ ,
Now from the trigonometric ratios we know that, $ \cos {45^0} = \dfrac{1}{{\sqrt 2 }} $ and $ \sin {45^0} = \dfrac{1}{{\sqrt 2 }} $ ,
Now substituting the cos and sin values we get,
\[ \Rightarrow v = \left( 1 \right)\dfrac{1}{{\sqrt 2 }}i + \left( 1 \right)\dfrac{1}{{\sqrt 2 }}j\],
Now simplifying we get,
$ \Rightarrow v = \dfrac{1}{{\sqrt 2 }}i + \dfrac{1}{{\sqrt 2 }}j $ ,
So, the component form is equal to $ v = \dfrac{1}{{\sqrt 2 }}i + \dfrac{1}{{\sqrt 2 }}j $ .
This can be represented using the graph,
The component form can also be written in the form $ \left( {\dfrac{1}{{\sqrt 2 }},\dfrac{1}{{\sqrt 2 }}} \right) $ .
$ \therefore $ The component form of the vector magnitude 1 and the angle it makes with the positive $ x $ -axis is $ \theta = {45^o} $ is equal to $ v = \dfrac{1}{{\sqrt 2 }}i + \dfrac{1}{{\sqrt 2 }}j $ , the component form can also be written as,\[\left( {\dfrac{1}{{\sqrt 2 }},\dfrac{1}{{\sqrt 2 }}} \right)\].
Note: For resolving a vector into its components, we can use the following formulas:
Consider $ a $ to be the magnitude of the vector $ \overrightarrow a $ and $ \theta $ to be the angle that is formed by the vector along the $ x $ -axis or to be the direction of the given vector. Then we will get,
\[{a_x} = \left| a \right| \times \cos \theta \]and $ {a_y} = \left| a \right| \times \sin \theta $ ,
Where $ {a_x} $ is called the magnitude of the $ x $ -component of the given vector $ \overrightarrow a $ ,
Where $ {a_y} $ is called the magnitude of the $ y $ -component of the given vector $ \overrightarrow a $ .
And the vector form is written as, $ \overrightarrow a = \left( {{a_x},{a_y}} \right) = \left| a \right|\cos \theta i + \left| a \right|\sin \theta j $ .
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
When people say No pun intended what does that mea class 8 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name the states which share their boundary with Indias class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give an account of the Northern Plains of India class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Advantages and disadvantages of science
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Bimbisara was the founder of dynasty A Nanda B Haryanka class 6 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
10 examples of evaporation in daily life with explanations
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you graph the function fx 4x class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)