Answer
There can be many vectors orthogonal to a given vector $\mathbf{v}$. Two possible answers are $\mathbf{u}=5\mathbf{i}+2\mathbf{j}$ and $\mathbf{u}=10\mathbf{i}+4\mathbf{j}$.
Work Step by Step
Let the vector orthogonal to $\mathbf{v}$ be $\mathbf{u}$. And it is expressed as
$\mathbf{u}=a\mathbf{i}+b\mathbf{j}$ …… (1)
For $\mathbf{u}$ and $\mathbf{v}$ to be orthogonal, their dot product must be zero, therefore
$\mathbf{u}\cdot \mathbf{v}=-2a+5b$ …… (2)
Keeping the dot product obtained in equation (2) as zero, we get
$5b=2a$ …… (3)
Equation (2) is a relation between $b$ and $a$ ; for different sets of $b$ and $a$, there will be different vectors orthogonal to $\mathbf{v}$.
Let us put $a=5$ in equation (3) to get
$\begin{align}
& a=5 \\
& b=2
\end{align}$
So, substitute these values of $b$ and $a$ in equation (1) to get
$\mathbf{u}=5\mathbf{i}+2\mathbf{j}$
Hence,
The vector $\mathbf{u}=5\mathbf{i}+2\mathbf{j}$ is orthogonal to $\mathbf{v}=-2\mathbf{i}+5\mathbf{j}$.
Let us put $a=10$ in equation (3) to get
$\begin{align}
& a=10 \\
& b=4
\end{align}$
So, substitute these values of $b$ and $a$ in equation (1) to get
$\mathbf{u}=10\mathbf{i}+4\mathbf{j}$
Hence,
The vector $\mathbf{u}=10\mathbf{i}+4\mathbf{j}$ is orthogonal to $\mathbf{v}=-2\mathbf{i}+5\mathbf{j}$.
