If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# Challenging complex numbers problem (1 of 3)

## Video transcript

let z1 and z2 be two distinct complex numbers and let Z equal and they say it's 1 minus T times z1 plus T times z2 for some real number with T being between 0 & 1 and they say if the argument W denotes the principal argument of a nonzero complex number W then and we know the argument is the angle of kind of the position vector specifying the complex number of the complex plane and the real and the real axis let me just draw that so if we do an Argand diagram this is the imaginary axis and this is the real axis if this is our complex number if that is Z its argument the argument of Z is going to be this angle right over here so Phi is equal to the argument the argument of Z that's all they're telling us in the second part now let's go through each of these and see are true and just to be clear this is one of those problems where more than one of these choices might be correct so let's see if they're true so let's first figure out what the magnitude of Z minus z1 is so the magnitude of Z minus z1 is going to be the magnitude of so let me write this is going to be the magnitude of let's to do the Z part first let me write in the same colors so if I would just focus on the Z first Z is this thing up here and let me distribute the z1 so it's Z 1 Z 1 minus T z1 minus T Z 1 and then plus T Z 2 plus T z2 and from that we want to subtract z1 so from that we want to subtract Z 1 so minus z1 and that and that cancel out that and that cancel out and let's see we have we can factor out a T over here so this is going to be equal to the magnitude of let's factor out a T times z2 minus z1 z2 minus minus z1 so that is the magnitude of Z minus z1 this first term over here let's figure out magnitude of Z - Z - the magnitude of Z - I'm gonna color-coded Z - Z 2 is equal to the magnitude well Z is just this thing up here I'm just write it out so it is 1 minus T times Z 1 plus T times Z 2 that's Z and from that we want to subtract Z 2 so minus minus Z 2 so what can we do over here we can take a we can take a t minus 1 out from here we can factor out the Z 2 out of these two terms so this is going to be equal to the magnitude of 1 minus T times z1 1 minus T times z1 that's this term right over here plus let me do it in another color plus t minus 1 plus t minus 1 t minus 1 times Z 2 times z2 the magnitude of this part over here is this thing over there now let's see what happens when we add this thing to this thing that's what choice a is doing we're adding this thing to this thing let's see if we can simplify it so this becomes this becomes the absolute value or the magnitude of T times z2 minus z1 plus I'll do the whole thing here in magenta Plus this thing over here and this thing over here actually before I even write it out how can we simplify that well I have a 1 minus T and a t minus 1 I'll just write it out here but I'm going to change it up a little bit so this is equal to the magnitude of 1 minus T 1 minus T times z1 and so that I have a 1 minus T here I'm just gonna put a negative out front so minus 1 minus T I just swap them negative 1 minus T is the same thing as positive t minus 1 z 2 and then this thing over here since I have the 1 minus T out there this becomes the magnitude of I'm just factoring out the 1 - t1 minus T times Z 1 minus Z 2 Z 1 minus Z 2 so we have this thing let me copy and paste it so copy and paste we have this thing Plus this thing this is what this expression a has simplified to now let's see if we can simplify that even more let's see if we can simplify it even more so T is just a scalar T is just a scalar and T is between 0 and 1 they told us that over here T is between 0 & 1 so this is positive this right here is a positive value and then this right here is also going to be this right here is also going to be a positive value T is greater than 0 and it is less than 1 so this is also going to be a positive value and these are just scalars so this is going to be the same thing these are just scaling the magnitude so this is going to be the same thing these are positive values so this is going to be T times the absolute value I actually let me not skip a step this is going to be the same thing as the absolute value of T times the absolute value of z2 minus z1 because this is just scaling it plus the absolute value of 1 minus T plus the absolute value of 1 minus T times the absolute value of z1 minus z2 now let me be clear the absolute value of z1 minus z2 is going to be equal to the absolute value of z2 minus z1 these vectors are just pointing in different directions or these complex numbers one is just the negative of the other but their absolute values or their magnitudes are going to be the same so let me write z2 minus z1 the absolute value of z2 minus z1 there the reason why I'm doing it is so I get the same thing here and I can factor it out so the absolute value of z2 minus z1 is the same thing as the absolute value of z1 minus z2 because we're not it's just going in the other direction now what can we do here well the absolute value of T remember T is positive so this is just this is just going to be equal to T the absolute value of 1 minus T once again that's positive so that's just going to be 1 minus T so we can factor we can factor this business out so we're going to get T plus one minus plus one minus T times the absolute value times the absolute value of Z 2 minus Z 1 Z 2 minus Z 1 now the t's cancel out here we just get a 1 out front so this is just equal to the absolute value of Z 2 minus Z 1 so we see the choice a does work this Plus this does indeed equal that I'm going to leave you there and in the next video we're going to try out some of these other possibilities to see if they might also be true