Question

How do you find the unit vector in the direction of the given vector of $u=<0,-2>$?

Answer 1

This type of problem is based on the concept of vectors. First, we need express $u=<0,-2>$ into a vector equation, that is, $\vec{u}=0\vec{i}+\left( -2 \right)\vec{j}$. Then, we should find the magnitude of$\vec{u}$, that is, $\left| {\vec{u}} \right|$. And then substitute the value into the unit vector formula, that is, $\hat{u}=\dfrac{{\vec{u}}}{\left| {\vec{u}} \right|}$ where $\hat{u}$ is the unit vector in the direction of the given vector $u=<0,-2>$. Therefore, we get the required solution.

Complete step-by-step answer:
According to the question, we are asked to find the unit vector in the direction of the given vector $u=<0,-2>$.
We have been given the vector is $u=<0,-2>$. -----(1)

We first have to convert equation (1) into a vector equation.
$\Rightarrow \vec{u}=0\vec{i}+\left( -2 \right)\vec{j}$ ------(2)
where $\vec{i}$ is the x-component and $\vec{j}$ is the y-component of the given vector.
Now let us find the magnitude of $\vec{u}$.
We know that, for the vector $\vec{r}=x\vec{i}+y\vec{j}$, the magnitude is
$\left| {\vec{r}} \right|=\sqrt{{{x}^{2}}+{{y}^{2}}}$
Therefore the magnitude of vector u is
$\left| {\vec{u}} \right|=\sqrt{{{0}^{2}}+{{\left( -2 \right)}^{2}}}$
$\Rightarrow \left| {\vec{u}} \right|=\sqrt{0+4}$
On further simplifications, we get,
$\left| {\vec{u}} \right|=\sqrt{4}$
$\therefore \left| {\vec{u}} \right|=2$ ---------(3)
Let us now find the unit vector in the direction of the vector $u=<0,-2>$.
We know that $\hat{u}=\dfrac{{\vec{u}}}{\left| {\vec{u}} \right|}$ , where $\hat{u}$ is the unit vector. -------(4)
Substituting (2) and (3) in (4), we get
$\hat{u}=\dfrac{0\vec{i}+\left( -2 \right)\vec{j}}{2}$
$\Rightarrow \hat{u}=\dfrac{0}{2}\vec{i}+\dfrac{\left( -2 \right)}{2}\vec{j}$
$\therefore \hat{u}=0\vec{i}+\left( -1 \right)\vec{j}$
Therefore, the unit vector is $<0,-1>$.

Hence, the unit vector in the direction of the given vector of $u=<0,-2>$ is $<0,-1>$ .

Note: Whenever you get this type of problem, we should always try to convert the vector into vector function to get the final of the function which will be the required answer. We should avoid calculation mistakes based on sign conventions. We should not get confused by unit vector and vector function. Similarly, we can also find the unit vector of u without converting u into $\vec{u}$ and reduce the number of steps to solve this type of questions.