Question

Find the modulus and amplitude of $ - 2i $
a. $ |z| = 2;amp(z) = - \dfrac{{3\pi }}{2} $
b. $ |z| = 2i;amp(z) = \dfrac{\pi }{2} $
c. $ |z| = 2;amp(z) = \dfrac{\pi }{2} $
d. $ |z| = 2;amp(z) = - \dfrac{\pi }{2} $

Answer 1

Hint: The number in the question is of the form complex number that is $ z = x + iy $ . The modulus of the complex number is defined as the square root of the sum of the square of the real part and the imaginary part. The complex number $ z = x + iy $ where $ x = |z|\cos \theta $ and $ y = |z|\sin \theta $ the $ \theta $ is called the amplitude of a complex number.

Complete step-by-step answer:
Consider the given number $ - 2i $ . The number is a complex number which is of the form $ z = x + iy $ . We can write the above question as $ z = - 2i $ where $ x = 0 $ and $ y = - 2 $ . The modulus of the complex number is given by $ |z| = \sqrt {\operatorname{Re} {{(z)}^2} + \operatorname{Im} {{(z)}^2}} $ ,where real part is x and imaginary part is y. Hence the modulus of a complex number is $ \Rightarrow |z| = \sqrt {{{(0)}^2} + {{( - 2)}^2}} $
On simplification we have
 $
   \Rightarrow |z| = \sqrt {0 + 4} \\
   \Rightarrow |z| = \sqrt 4 \;
  $
Applying the square root, we have
 $ |z| = 2 $
Hence the modulus of the complex number $ - 2i $ is 2.
Now we have to find the amplitude of the complex number $ - 2i $
Consider the given question $ z = - 2i $ . The number is a complex number which is of the form $ z = x + iy $ . where $ x = 0 $ and $ y = - 2 $ . Where x is the real part and y is the imaginary part. We can write x and y has $ x = |z|\cos \theta $ and $ y = |z|\sin \theta $ .
 $ \Rightarrow \dfrac{y}{x} = \dfrac{{|z|\sin \theta }}{{|z|\cos \theta }} $
Cancelling the like terms, we have
 $ \Rightarrow \dfrac{y}{x} = \tan \theta $
And we can rewrite as, $ \Rightarrow \theta = {\tan ^{ - 1}}\left( {\dfrac{y}{x}} \right) $
 Therefore, the amplitude of a complex number is defined as $ \theta = {\tan ^{ - 1}}\left( {\dfrac{y}{x}} \right) $
By substituting the values to the formula, we have $ \Rightarrow \theta = {\tan ^{ - 1}}(\dfrac{{ - 2}}{0}) $
We know that any number divided by 0 is infinity
 $ \Rightarrow \theta = {\tan ^{ - 1}}( - \infty ) $
 $ \Rightarrow \theta = \dfrac{{ - \pi }}{2} $
Since the point $ - 2i $ lies on the negative half of the imaginary axis. So the amplitude is $ \dfrac{{ - \pi }}{2} $
Hence the amplitude of $ - 2i $ is $ \dfrac{{ - \pi }}{2} $
Therefore, the modulus and amplitude of $ - 2i $ is 2 and $ \dfrac{{ - \pi }}{2} $ respectively.
So, the correct answer is “Option D”.

Note: Since the given number is a complex number which contains the both real and imaginary part. We should identify the real part and imaginary part of the number and hence by using the formula we can determine the modulus and amplitude of the number.