Question

The minimum value of $\left| a+b\omega +c{{\omega }^{2}} \right|$ where $a,b,c$ are all not equal integers and $\omega \left( \ne 1 \right)$ is a cube root of unity, is

1. $\sqrt{3}$
2. $\dfrac{1}{2}$
3. $1$
4. $0$

Answer 1

In this type of question we have to use the concept of the cube root of unity. We know that the cube root of unity is represented by $\omega$. Also we know that there are three cube roots of unity namely $1,\omega ,{{\omega }^{2}}$ and their sum is equal to zero i.e. $1+\omega +{{\omega }^{2}}=0$.

Complete step-by-step solution:
Now we have to find the value of $\left| a+b\omega +c{{\omega }^{2}} \right|$ where $a,b,c$ are all not equal integers and $\omega \left( \ne 1 \right)$ is a cube root of unity
For this let us consider
$\Rightarrow z=\left| a+b\omega +c{{\omega }^{2}} \right|$
Now as we know that there are three cube roots of unity namely $1,\omega ,{{\omega }^{2}}$ and their sum is equal to zero i.e. $1+\omega +{{\omega }^{2}}=0$
$\Rightarrow {{\omega }^{2}}=-1-\omega$
By substituting this in above expression we get,

& \Rightarrow z=\left| a+b\omega -c-c\omega \right| \\\ & \Rightarrow z=\left| \left( a-c \right)+\left( b-c \right)\omega \right| \\\ \end{aligned}$$ $$\Rightarrow z\ge \left| a-c \right|+\left| \left( b-c \right)\omega \right|$$ Now if $$\omega =1$$ then we get $$\Rightarrow z=\left| a-c \right|+\left| \left( b-c \right) \right|$$ Now for $$z$$ to be minimum $$\Rightarrow \left| a-c \right|=0,\left| b-c \right|=0$$ $$\Rightarrow a=c,b=c$$ Which is not possible as we have given that $$a,b,c$$ are all not equal integers Thus for $$z$$ to be minimum we have $$\Rightarrow \text{If }\left| a-c \right|=0\text{ then }\left| b-c \right|=1$$ Hence, we can write $$\begin{aligned} & \Rightarrow z\ge \left| a-c \right|+\left| \left( b-c \right)\omega \right| \\\ & \Rightarrow z\ge 1 \\\ \end{aligned}$$ **Thus, option (3) is the correct option.** **Note:** In this question students must be familiar with the cube root of unity. Students have to note that to represent the cube root of unity we use the symbol $$\omega $$. The cube root of unity has three values out of which one value is 1 and the remaining two are the complex values which we denote as $$\omega $$ and $${{\omega }^{2}}$$.