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# Need help on this math question about magnitudes

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Let $z$ and $w$ be complex numbers satisfying $|z| = 5, |w| = 2,$ and $z\overline{w} = 6+8i.$ Then enter in the numbers $|z+w|^2, |zw|^2, |z-w|^2, \left| \dfrac{z}{w} \right|^2$below, in the order listed above. If any of these cannot be uniquely determined from the information given, enter in a question mark.

I got 49 for the first one, 100 for the second one, 9 for the 3rd one, and 6.25 for the 4th one. But it was wrong, so I don't know how to do this question.

Jun 28, 2020
edited by FearlessIsland3  Jun 28, 2020

#1
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Just use the formula for |z|^2.

Jun 28, 2020
#2
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Clean up your question. Put your LaTex in a LaTex box  (found on the ribbon)

Jun 29, 2020
#3
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$$z\overline{w} = 6 + 8i\\ \overline{z\overline{w}} = \overline{6+8i}\\ \overline{z}w = 6 - 8i$$

And then, we know that for any complex number z and w, $$z\overline{z} = |z|^2$$ and $$|z||w| = |zw|$$.

$$\quad|zw|^2 \\= |z|^2 |w|^2 \\= z\overline{z}w\overline{w} \\= (z\overline{w})(w\overline z) \\= (6 + 8i)(6 - 8i) \\= 10$$

$$\quad|z + w|^2 \\= (z + w)\left(\overline{z + w}\right) \\= (z + w)\left(\overline z + \overline w\right)\\ = z\overline z + w\overline w + z\overline w + w\overline z \\= |z|^2 + |w|^2 + z\overline w + \overline zw \\= 5^2 + 2^2 + (6 + 8i) + (6- 8i) \\= 41$$

I will leave the remaining ones to you.

Jun 29, 2020
edited by MaxWong  Jun 29, 2020
edited by MaxWong  Jun 29, 2020