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  • Find the unit vector in the direction of sum of vectors a =2i -j + k and b= 2j+k

Find the unit vector in the direction of sum of vectors \overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}} \quad \text { and } \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{j}}+\hat{\mathrm{k}}

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We have,

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Since, unit vector is needed to be found in the direction of the sum of vectors https://gradeup-question-images.grdp.co/liveData/PROJ28891/1554370408717444.png  and https://gradeup-question-images.grdp.co/liveData/PROJ28891/1554370409458743.png .
So, add vectors https://gradeup-question-images.grdp.co/liveData/PROJ28891/1554370410214789.png  and https://gradeup-question-images.grdp.co/liveData/PROJ28891/1554370410957395.png .
Let,
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Substituting the values of vectors https://gradeup-question-images.grdp.co/liveData/PROJ28891/1554370412468682.png  and https://gradeup-question-images.grdp.co/liveData/PROJ28891/1554370413235486.png .

\\ \Rightarrow \vec{c}=(2 \hat{\imath}-\hat{\jmath}+\hat{\mathrm{k}})+(2 \hat{\jmath}+\hat{\mathrm{k}}) \\ \Rightarrow \overrightarrow{\mathrm{c}}=2 \hat{\imath}-\hat{\jmath}+\hat{\mathrm{k}}+2 \hat{\jmath}+\hat{\mathrm{k}} \\ \Rightarrow \overrightarrow{\mathrm{c}}=2 \hat{\imath}-\hat{\jmath}+2 \hat{\jmath}+\hat{\mathrm{k}}+\hat{\mathrm{k}} \\ \Rightarrow \overrightarrow{\mathrm{c}}=2 \hat{\imath}+\hat{\jmath}+2 \hat{\mathrm{k}}

We know that a unit vector in a normed vector space is a vector (often a spatial vector) of length 1.

To find a unit vector with the same direction as a given vector, we divide by the magnitude of the vector.

For finding unit vector, we have the formula:

\\ \begin{aligned} &\hat{c}=\frac{\overrightarrow{\mathrm{c}}}{|\overrightarrow{\mathrm{c}}|}\\ &\text { Substitute the value of } \overrightarrow{\mathrm{c}} \text { . }\\ &\Rightarrow \hat{c}=\frac{2 \hat{\imath}+\hat{\jmath}+2 \hat{\mathrm{k}}}{|2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+2 \hat{\mathrm{k}}|}\\ &\text { Here, }|2 \hat{\imath}+\hat{\jmath}+2 \hat{\mathrm{k}}|=\sqrt{2^{2}+1^{2}+2^{2}} \end{aligned}

\\ \Rightarrow \hat{c}=\frac{2 \hat{\imath}+\hat{\jmath}+2 \hat{\mathrm{k}}}{\sqrt{2^{2}+1^{2}+2^{2}}} \\ \Rightarrow \hat{\mathrm{c}}=\frac{2 \hat{\imath}+\hat{\jmath}+2 \hat{\mathrm{k}}}{\sqrt{4+1+4}} \\ \Rightarrow \hat{c}=\frac{2 \hat{\imath}+\hat{\jmath}+2 \hat{\mathrm{k}}}{\sqrt{9}} \\ \Rightarrow \hat{c}=\frac{2 \hat{\imath}+\hat{\jmath}+2 \hat{\mathrm{k}}}{3}

Thus, unit vector in the direction of sum of vectors \vec{a}{\text { and }} \vec{b}   is \frac{2 \hat{i}+\hat{j}+2 \hat{k}}{3} .

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