If a = i + j , b = j + k and c = k + i , write unit vectors paral… - Vidyadip

Question

If $\vec{\text{a}}=\hat{\text{i}}+\hat{\text{j}},\ \vec{\text{b}}=\hat{\text{j}}+\hat{\text{k}}$ and $\vec{\text{c}}=\hat{\text{k}}+\hat{\text{i}}$, write unit vectors parallel to $\vec{\text{a}}+\vec{\text{b}}-2\vec{\text{c}}$.

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Answer

Given: $\vec{\text{a}}=\hat{\text{i}}+\hat{\text{j}},\ \vec{\text{b}}=\hat{\text{j}}+\hat{\text{k}},\ \vec{\text{c}}=\hat{\text{k}}+\hat{\text{i}}$Now, $\vec{\text{a}}+\vec{\text{b}}-2\vec{\text{c}}=\hat{\text{i}}+\hat{\text{j}}+\hat{\text{j}}+\hat{\text{k}}-2\hat{\text{k}}-2\hat{\text{i}}$
$=-\hat{\text{i}}+2\hat{\text{j}}-\hat{\text{k}}$ Unit vector parallel to $\vec{\text{a}}+\vec{\text{b}}-2\vec{\text{c}}=\frac{-\hat{\text{i}}+2\hat{\text{j}}-\hat{\text{k}}}{\sqrt{(-1)^2+2^2+(-1)^2}}$ $$$=\frac{-\hat{\text{i}}+2\hat{\text{j}}-\hat{\text{k}}}{\sqrt6}$

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