Write a vector parallel to the vector 2 i-j whose magnitude is 5 … - Vidyadip

Question

Write a vector parallel to the vector $2 \hat{i}-\hat{j}$ whose magnitude is $5$ units.

✓

Answer

Let the vector parallel to the vector $2 \hat{i}-\hat{j}$ be $a_1 \hat{i}+a_2 \hat{j}$. Given that
$
\begin{aligned}
\left|a_1 \hat{i}+a_2 \hat{j}\right| & =5 \\
\because \quad|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
\end{aligned}
$
For being parallel :
$
\begin{aligned}
\frac{a_1}{2} & =\frac{a_2}{-1}=k \\
a_1 & =2 k \\
a_2 & =-k
\end{aligned}
$
Putting the values in equation (1),
$
\begin{array}{l}
(2 k)^2+(-k)^2=25 \\
\Rightarrow \quad 4 k^2+k^2=25 \\
5 k^2=25 \\
k=\sqrt{5} \\
\therefore a_1=2 \sqrt{5} \text { and } a_2=-\sqrt{5}
\end{array}
$
Hence the parallel vector will be $2 \sqrt{5} \hat{i}-\sqrt{5} \hat{j}$.

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