Question

If $z=(3+4i)/i$ , then find $|z|$​

Answer 1

Answer:

z = 7

Step-by-step explanation:

z = 3 + 4i/i

~Cancel out

z = 3 + 4

~Simplify

z = 7

Best of Luck!

Answer 2

ANSWER :

• $\: \: \: \bigstar \: \: \: \: \large{ \bold{ If \: \: \dfrac{3 + 4i}{i} , \: then \: \: |z| = 5 \: }}$

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SOLUTION :

❒ Given :-

• $\: \: \large{ \rm{z = \dfrac{ \: 3 + 4i \: }{i}}}$

❒ To Find :-

• $\: \: \: \: \: \: \: \large{ \rm{ |z| = \: ?}}$

❒ Calculation :-

It is given that,

• $\bigstar \: \: \: \sf{z = \dfrac{ \: 3 + 4i \: }{i}}$

This can be also written as,

• $\sf {\implies \: z = (3 + 4i) \times \dfrac{1}{i}}$

We know that,

• $\: \: \: \: \: \underline{ \boxed{ \bold{ \: \: \dfrac{1}{i} = - i }}}$

Hence,

• $\sf{\implies \: z = (3 + 4i) \times ( - i)}$

Multiplying (3 + 4i) with ( - i), we get,

• $\sf {\implies \: z = 3 \times ( - i) + 4i \times ( - i)} \\ \\ \sf {\implies \: z = - 3i - 4 i {}^{2}} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

We know that,

• $\: \: \: \: \: \underline{ \boxed{ \bold{ \: \: i {}^{2} = - 1 \: \: }}}$

Hence,

• $\sf \implies \: z = - 3i - 4 \times ( - 1) \\ \\ \sf \implies \: z = - 3i + 4 \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf{ \therefore \: \: \underline{ \: z = 4 - 3i \: }} \: \: \: \: \: \:$

We know that,

• $\: \: \: \: \bold{If \: \: z = a + ib \: \: is \: \: a \: \: complex \: \: number, \: then} \\ \underline{ \boxed{ \bold{ \: \: |z| = \: \sqrt{a {}^{2} +b {}^{2}} \: \: }}} \: \:$

Here,

• z = 4 - 3i

So,

• a = 4

• b = - 3

Hence,

• $\: \: \: \: \bigstar \: \: \sf{ |z| = \sqrt{(4) {}^{2} + ( - 3) {}^{2} }}$

$\sf{ \implies \: |z| = \sqrt{16 + 9}} \\ \\ \sf{ \implies \: |z| = \sqrt{25}} \: \: \: \: \: \: \: \\ \\ \sf{ \therefore \: \: \underline{ \: |z| = 5 \: }} \: \: \: \: \: \: \: \: \: \:$