MHB If z = -3+4i....(related vectors)

  • Thread starter Thread starter Raerin
  • Start date Start date
  • Tags Tags
    Vectors
AI Thread Summary
The discussion revolves around the complex number z = -3 + 4i and its related calculations. Participants clarify that the notation with a line above z indicates its conjugate, while the magnitude |z| is calculated as 5. The conversation includes detailed steps for finding the reciprocal of z, resulting in -3/25 - 4i/25. There is also a focus on understanding the implications of dividing by z versus its magnitude. Overall, the thread effectively addresses the calculations and concepts related to complex numbers.
Raerin
Messages
46
Reaction score
0
If z = -3+4i, determine the following related complex numbers

a) vector z
b) 3(vector z)
c) 1/z
d) 1/(vector z)
e) |z|
f) |vector z|
g) (vector z)/(|z|^2)

I'm not sure if it's a vector, but the z has a short line above it when I say "vector z."
 
Mathematics news on Phys.org
Re: If z = -3+4i...

The line over a complex number refers to its conjugate. If $z=a+bi$, then $\overline{z}=a-bi$. Are you familiar with the other notations?
 
Re: If z = -3+4i...

MarkFL said:
The line over a complex number refers to its conjugate. If $z=a+bi$, then $\overline{z}=a-bi$. Are you familiar with the other notations?
Nope :( but |z| refers to the length of it? So it'll be the radius of a circle?
 
Re: If z = -3+4i...

Yes, $|z|$ refers to the magnitude, which is given by:

$$|z|=\sqrt{a^2+b^2}$$

So, what do you find for the magnitude of the given complex number?
 
Re: If z = -3+4i...

MarkFL said:
Yes, $|z|$ refers to the magnitude, which is given by:

$$|z|=\sqrt{a^2+b^2}$$

So, what do you find for the magnitude of the given complex number?

Ahh, I see, so |z| would be 5.

Also, when 1 is divided by z, is it the same as dividing 1 with...5, let's say?
 
Re: If z = -3+4i...

Yes, correct on both counts. (Sun)

So, what about parts b) and c)?
 
Re: If z = -3+4i...

MarkFL said:
Yes, correct on both counts. (Sun)

So, what about parts b) and c)?

Okay, I understand everything now. Thanks for your help!
 
Re: If z = -3+4i...

Hi Raerin, welcome to MHB! :)

Raerin said:
Also, when 1 is divided by z, is it the same as dividing 1 with...5, let's say?

I'd like to add a bit of nuance here.

\begin{aligned}
\frac 1 z &= \frac 1 {-3+4i} \\
&= \frac 1 {-3+4i} \cdot \frac {-3-4i} {-3-4i} \\
&= \frac {-3-4i} {(-3+4i)(-3-4i)} \\
&= \frac {-3-4i}{(-3)^2-(4i)^2} \\
&= \frac {-3-4i}{9+16} \\
&= \frac 1 {25} (-3-4i)
\end{aligned}
 
Re: If z = -3+4i...

I like Serena said:
Hi Raerin, welcome to MHB! :)
I'd like to add a bit of nuance here.

\begin{aligned}
\frac 1 z &= \frac 1 {-3+4i} \\
&= \frac 1 {-3+4i} \cdot \frac {-3-4i} {-3-4i} \\
&= \frac {-3+4i} {(-3+4i)(-3-4i)} \\
&= \frac {-3-4i}{(-3)^2-(4i)^2} \\
&= \frac {-3-4i}{9+16} \\
&= \frac 1 {25} (-3-4i)
\end{aligned}

Yes, I was mistakenly referring to $$\frac{1}{|z|}$$. Good catch! (Yes)
 
Back
Top