Question

If U = 2 - i, V = - 2+ i , then Re (UV / V bar) is​

Answer 1

Complex Numbers

Given: $U=2-i,\:V=-2+i$

To find: $Re(\frac{UV}{\overline{V}})$

SOLUTION:

• Now, $UV=(2-i)(-2+i)$

• $=-4+2i+2i-i^{2}$

• $=-4+4i+1$, since $i=\sqrt{-1}$

• $=-3+4i$

• and $\overline{V}$ is the conjugate of $V$; so,

• $\quad \overline{V}=-2-i$

• $\therefore \frac{UV}{\overline{V}}$

• $=\frac{-3+4i}{-2-i}$

• $=\frac{3-4i}{2+i}$

• $=\frac{(3-4i)(2-i)}{(2+i)(2-i)}$,

• where we have multiplied both the numerator and the denominator by the conjugate of the denominator

• $=\frac{6-3i-8i+4i^{2}}{4-i^{2}}$

• $=\frac{6-11i-4}{4+1}$

• $=\frac{2-11i}{5}$

• $\Rightarrow\frac{UV}{\overline{V}}=\frac{2}{5}-\frac{11}{5}i$

Answer: $\therefore Re(\frac{UV}{\overline{V}})=\frac{2}{5}$.