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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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