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Vector representations Quiz 2

mathematics

 The unit vector of $\text{12}\widehat{i}+39\widehat{j}\text{ }$ is 

 The unit vector of $a\widehat{i}+b\widehat{j}$ is equal to $\dfrac{a\widehat{i}+b\widehat{j}}{|a\widehat{i}+b\widehat{j}|}$.

  The unit vector of $a\widehat{i}+b\widehat{j}$ is equal to $\dfrac{a\widehat{i}+b\widehat{j}}{|a\widehat{i}+b\widehat{j}|}$ $ \\ $ The unit vector of $\text{12}\widehat{i}+39\widehat{j}\text{ }=\dfrac{\text{12}\widehat{i}+39\widehat{j}\text{ }}{|\text{12}\widehat{i}+39\widehat{j}\text{ }|}=\dfrac{\text{12}\widehat{i}+39\widehat{j}\text{ }}{\sqrt{1523}}$

 Which of the following statements is/are correct?

 The x-component of the vector $a\widehat{i}+b\widehat{j}$ is equal to a. The resultant vector of two vectors is equal to the sum of two vectors.

  The magnitude of $a\widehat{i}+b\widehat{j}$ is equal to $\sqrt{{{a}^{2}}+{{b}^{2}}}$ $ \\ $ The x-component of $177$ $\widehat{i}+273\widehat{j}$ is equal to $177$ $ \\ $ The resultant vector of $\widehat{i}+28\widehat{j}$ and $12$ $\widehat{i}+39\widehat{j}$ is equal to $ \\ \Rightarrow R=\left( \widehat{i}+28\widehat{j} \right)+\left( 12\widehat{i}+39\widehat{j} \right) \\  \Rightarrow R=\text{13}\widehat{i}+67\widehat{j}$

 A stone is thrown with a velocity $145m/s$ which makes an angle  ${{37}^{\circ }}$ to the ground. Then find the velocity vector through which the stone travels?

 If a vector $\widehat{v}$ makes an angle $\theta $ with x-axis, then $\widehat{v}=\left( v\cos \theta  \right)\widehat{i}+\left( v\sin \theta  \right)\widehat{j}$. By using this condition, find the required velocity vector.

If a vector $\widehat{v}$ makes an angle $\theta $ with x-axis, then $\widehat{v}=\left( v\cos \theta  \right)\widehat{i}+\left( v\sin \theta  \right)\widehat{j}$ $ \\ $ It is given that stone is thrown with a velocity $145m/s$ $ \\ v\cos \theta =145\left( \dfrac{4}{5} \right)=116 \\  v\sin \theta =145\left( \dfrac{3}{5} \right)=77 \\ $ So, $ \\ \widehat{v}=116\widehat{i}+77\widehat{j}$

 The magnitude of the vector $24\widehat{i}+18\widehat{j}$ is equal to a. The component of the vector $116\widehat{i}+77\widehat{j}$ on the x-axis is equal to b. Find the value of $\dfrac{|b+4|}{a}$.

 The magnitude of the vector $a\widehat{i}+b\widehat{j}$ is equal to $\sqrt{{{a}^{2}}+{{b}^{2}}}$.

 The magnitude of the vector $a\widehat{i}+b\widehat{j}$ is equal to $\sqrt{{{a}^{2}}+{{b}^{2}}}$ $ \\ \Rightarrow a=\sqrt{{{24}^{2}}+{{18}^{2}}}=30 \\ $ The component of the vector $116\widehat{i}+77\widehat{j}$ on x-axis is equal to $116$ $ \\\Rightarrow b=116 \\ \dfrac{|b+4|}{a}=\dfrac{|116+4|}{30}=4$

 The resultant vector of  $87\widehat{i}+58\widehat{j}$  and $58\widehat{i}+87\widehat{j}$ is $p\widehat{i}+q\widehat{j}$. The angle between $p\widehat{i}+q\widehat{j}$ and $\widehat{j}$ is $\theta $. Now find the vector which makes the angle $\alpha $ with x-axis and has a magnitude equal to $p\widehat{i}+q\widehat{j}$ where $\alpha =\theta -8$.

 The resultant vector of two vectors is equal to the sum of two vectors. The magnitude of the vector $a\widehat{i}+b\widehat{j}$ is equal to $\sqrt{{{a}^{2}}+{{b}^{2}}}$.

 Resultant vector of $87\widehat{i}+58\widehat{j}$  and $58\widehat{i}+87\widehat{j}=145\widehat{i}+145\widehat{j}$ $ \\ $ Let the angle between $145\widehat{i}+145\widehat{j}$ and $\widehat{j}$ be $\theta $ $ \\ \cos \theta =\dfrac{\left( 145\widehat{i}+145\widehat{j} \right)\left( {\widehat{j}} \right)}{145\sqrt{2}}=\dfrac{1}{\sqrt{2}}\Rightarrow \theta ={{45}^{\circ }} \\ \alpha =\theta -8=45-8={{37}^{\circ }} \\ $ Magnitude of $145\widehat{i}+145\widehat{j}=\sqrt{{{145}^{2}}+{{145}^{2}}}=145\sqrt{2}$ $ \\ $ Vector whose magnitude is equal to $145\sqrt{2}$ and makes an angle ${{37}^{\circ }}$ is, $ \\ 145\sqrt{2}\cos {{37}^{\circ }}+145\sqrt{2}\sin {{37}^{\circ }}=116\sqrt{2}\widehat{i}+77\sqrt{2}\widehat{j}$