Gujarat BoardEnglish MediumSTD 12 ScienceMathsVector Algebra2 Marks
Question
Write a vector parallel to the vector $2 \hat{i}-\hat{j}$ whose magnitude is $5$ units.
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Answer
Let the vector parallel to the vector $2 \hat{i}-\hat{j}$ be $a_1 \hat{i}+a_2 \hat{j}$. Given that
$\left|a_1 \hat{i}+a_2 \hat{j}\right| =5$
$\because |2 \hat{i}-\hat{j}| =\sqrt{(2)^2+(-1)^2}=\sqrt{5}$
$\sqrt{a_1^2+a_2^2} =5$
$a_1^2+a_2^2 =25$
For being parallel :
$\frac{a_1}{2} =\frac{a_2}{-1}=k$
$a_1 =2 k$
$a_2 =-k$
Putting the values in equation $(1),$
$(2 k)^2+(-k)^2=25$
$\Rightarrow 4 k^2+k^2=25$
$5 k^2=25$
$k=\sqrt{5}$
$\therefore a_1=2 \sqrt{5}$ and $a_2=-\sqrt{5}$
Hence the parallel vector will be $2 \sqrt{5} \hat{i}-\sqrt{5} \hat{j}$.
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