Question

# Find a vector of magnitude 5 units, and parallel to the resultant of the vectors.$$\overrightarrow{a} = 2 \hat{i} + 3 \hat{j} -\hat{k}$$ and $$\overrightarrow{b}=\hat{i} - 2 \hat{j} + \hat{k}$$

Solution

## Given two vectors are $$\overrightarrow {a} = 2 \hat { i} + 3 \hat { j } - \hat {k}$$ and $$\overrightarrow {b} = \hat {i} - 2 \hat {j} + \hat {k}$$If $$\overrightarrow {c}$$ is the resultant vector of $$\overrightarrow {a}$$ and $$\overrightarrow {b} = ( 2 \hat {i} + 3 \hat {j} - \hat {k} ) + ( \hat {i} - 2 \hat {j} + \hat {k}) = 3 \hat {i} + \hat { j} + 0. \hat {k}$$Now, a vector having magnitude $$5$$ and parallel to $$\overrightarrow {c}$$ is given by$$\frac {5 \overrightarrow {c}}{| \overrightarrow{c} |} = \dfrac {5( 3 \hat {i} + \hat {j} 0 \hat {k})}{\sqrt { 3^2 +1^2 +0^2}} = \dfrac {15}{\sqrt {10}} \hat {i} + \dfrac {5}{\sqrt {10} } \hat {j}$$It is required vector.[ Note : A vector having magnitude $$I$$ and parallel to $$\overrightarrow {a}$$ is given by $$I \dfrac { \overrightarrow {a} }{ | \overrightarrow {a} |} . ]$$Physics

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