Video Transcript
Given that π§ sub one is equal to two root three plus two π and π§ sub two is equal to negative two minus two root three π, find π§ sub one multiplied by π§ sub two, giving your answer in exponential form.
In this question, weβre given two complex numbers in the form π₯ plus π¦π, where π₯ is the real part and π¦ is the imaginary part of the complex number. Weβre asked to calculate the product of the two complex numbers π§ sub one and π§ sub two. We can do this by distributing the parentheses using the FOIL method. Multiplying the first terms gives us negative four root three. Multiplying the outer or outside terms gives us negative 12π. This is because two multiplied by negative two is negative four. Root three multiplied by root three is equal to three. Finally, negative four multiplied by three is equal to negative 12.
Multiplying the inner or inside terms gives us negative four π. Finally, multiplying the last terms gives us negative four root three π squared. We recall that when dealing with complex numbers, π squared is equal to negative one. This means that negative four root three π squared is equal to positive four root three. We can now collect like terms. The real terms cancel as negative four root three plus four root three is equal to zero. π§ one multiplied by π§ two is therefore equal to negative 16π.
We recall that any complex number can be written in exponential form such that π§ is equal to π multiplied by π to the ππ, where π is the modulus of the complex number and π is its argument. The modulus of a complex number is equal to the square root of π₯ squared plus π¦ squared, where π₯ and π¦ are the real and imaginary parts, respectively. The argument π is equal to the inverse tan of π¦ over π₯. As there is no real part to the complex number negative 16π, then π₯ is equal to zero, and π¦ is equal to negative 16. π is therefore equal to the square root of zero squared plus negative 16 squared. This is equal to 16.
π is equal to the inverse tan of negative 16 over zero. As the denominator is equal to zero, this will be undefined. We are looking for the value of π where tan π is undefined. This occurs at π over two plus ππ. In order to work out the correct value of π for this question, we will clear some space and draw the Argand diagram.
The complex number π§ one π§ two had a real value equal to zero and an imaginary value equal to negative 16. This lies at the point which separates the third and fourth quadrants. We know that this corresponds to the angle three π over two. The value of the argument π is three π over two. We can therefore conclude that the complex number π§ one π§ two written in exponential form is equal to 16π to the three π over two π.