If z1+z2+z3=0r then |z2-z3|^2+|z3-z1|^2+ |z1-z2 ^2 equals (a) (1)/(3)|z1|^2+.2 j2

step 1 I have a given here Z1 plus Z2 plus Z3 equal to 0. Here we have that one that's three represent

If z1+z2+z3=0r then |z2-z3|^2+|z3-z1|^2+ |z1-z2...

$$ \begin{aligned} &\text { If } z_{1}+z_{2}+z_{3}=0_{r} \text { then }\left|z_{2}-z_{3}\right|^{2}+\left|z_{3}-z_{1}\right|^{2}+ \\ &\mid z_{1}-z_{2}{ }^{2} \text { equals } \end{aligned} $$ (a) $\frac{1}{3}\left|z_{1}\right|^{2}+\left.2 j_{2}\right|^{2}+2\left|z_{3}\right|^{2}$ (b) $\frac{2}{3}\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{3}\right|^{2}\right)$ (c) $2\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{3}\right|^{2}\right)$ (d) $3\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{3}\right|^{2}\right)$

$$
\begin{aligned}
&\text { If } z_{1}+z_{2}+z_{3}=0_{r} \text { then }\left|z_{2}-z_{3}\right|^{2}+\left|z_{3}-z_{1}\right|^{2}+ \\
&\mid z_{1}-z_{2}{ }^{2} \text { equals }
\end{aligned}
$$
(a) $\frac{1}{3}\left|z_{1}\right|^{2}+\left.2 j_{2}\right|^{2}+2\left|z_{3}\right|^{2}$
(b) $\frac{2}{3}\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{3}\right|^{2}\right)$
(c) $2\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{3}\right|^{2}\right)$
(d) $3\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{3}\right|^{2}\right)$

Key Concepts

Complex Norm

The complex norm provides a measure of the length or magnitude of a complex number. When treating complex numbers as vectors in the plane, the norm is defined as the square root of the sum of the squares of the real and imaginary parts. The square of the norm, |z|², is frequently used in algebraic manipulations and geometric interpretations, such as computing distances or energies, and is fundamental in analysis and geometry on complex domains.

Pairwise Distance Sum Formula

This is an identity that relates the sum of the squares of the differences (or distances) between pairs of points to the sum of the individual squared norms of those points. In an n-point configuration, the formula ?|z_i - z_j|² = n?|z_i|² - |?z_i|² provides a powerful tool to simplify expressions involving pairwise differences by connecting them to the points’ magnitudes and their collective sum.

Centroid (Zero Sum Condition)

The condition that the sum of the complex numbers is zero indicates that their centroid (or center of mass) is at the origin. This property greatly simplifies many expressions. For instance, in the pairwise distance sum formula, if ?z_i = 0, then the formula reduces to n?|z_i|². This simplification is a key insight in problems where the vanishing sum of vectors leads to easier computation of quantities like the total squared distances between points.

Transcript

00:01 I have a given here z1 plus z2 plus z3 equal to 0.

00:05 Here we have that one that's three represent complex numbers.

00:08 We're going to find out what the relation bring out to be.

00:12 So if we just apply here, you have a basic formula we apply here.

00:16 For we have two complex numbers, a and b, a negative b, module of square that's given as mod of a square plus model of b square, negative a, b conjugate, negative a conjugate, b.

00:28 This formula now we're going to apply here.

00:30 So we get mod z2 square plus more z three square, negative z two, z, z3 conjugate, negative z2 conjugate, z3.

00:45 Then we have more z3 square plus more z1 square.

00:51 Then we have negative z3, z1 conjugate, negative z3 conjugate, z1.

00:58 Next we have mode z1 square.

01:01 Plus more z2 square next we have negative z1 z2 conjugate negative z z1 conjugate z2 so this i've got here now if i simplify this so we get it as here more z2 square here more that 2 square here more z2 square plus more z 3 square this we have got.

01:34 The next we're given here that is negative z2, z3 conjugate, and so what we can do here, we're given here this relation that is z1 plus that triple that 3 equal to 0.

01:56 Before that we rearrange here, so here we have it is that negative that 3 conjugate z1, here we have negative z1, that 2 conjugate.

02:04 We factor out that 1 here.

02:06 Similarly, we factor out z2 and z2 and a factor out that 3 and that 3...

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