Video Transcript
Consider the complex number 𝑧 is equal to five plus three 𝑖. If 𝑖 multiplied by 𝑧 is represented on an Argand diagram by the point 𝐴, in which quadrant of the Argand plane does 𝐴 lie?
We begin by sketching an Argand diagram. The 𝑥-axis represents the real part of our complex number, and the 𝑦-axis represents the imaginary part. The four quadrants can be labeled using Roman numerals as shown. The first quadrant is the top right, where both the real and imaginary parts are positive. The second quadrant is the top left, the third quadrant the bottom left, and the fourth quadrant the bottom right.
In this question, we are told that the complex number 𝑧 is equal to five plus three 𝑖. On the Argand diagram, this will be located at the point with coordinates five, three. The real part of our complex number is equal to five, and the imaginary part is three. As both of these are positive, our point lies in the first quadrant.
We are interested in the complex number 𝑖𝑧. This means we need to multiply five plus three 𝑖 by 𝑖. Distributing the parentheses here gives us five 𝑖 plus three 𝑖 squared. We recall that 𝑖 squared is equal to negative one. This means that 𝑖𝑧 is equal to five 𝑖 plus three multiplied by negative one. This is equal to five 𝑖 minus three or negative three plus five 𝑖. We are told that this is denoted on the Argand diagram by point 𝐴. As the real part of our complex number is negative three and the imaginary part is positive five, point 𝐴 has coordinates negative three, five.
We can therefore conclude that the complex number 𝑖𝑧, which is equal to negative three plus five 𝑖, lies in the second quadrant of the Argand plane.
