Question

If Z₁=3-2i
Z2=-2+3i
Z3=4+2i
Evaluate Z₂ Z3/Z₁

Answer 1

32/13

Explanation:

To evaluate the expression Z₂ * Z3 / Z₁, we first need to find the complex numbers Z₂ * Z3 and Z₁.

Z₂ * Z3 = -2 + 3i * 4 + 2i = (-8 + 14i) + (-6 + 6i) = -8 + 8i

Z₁ = 3 - 2i

So, Z₂ * Z3 / Z₁ = (-8 + 8i) / (3 - 2i)

We can simplify this expression by multiplying the numerator and denominator by the complex conjugate of the denominator, which is (3 + 2i).

Z₂ * Z3 / Z₁ = (-8 + 8i) / (3 - 2i) * (3 + 2i) / (3 + 2i) = (-8 + 8i) * (3 + 2i) / (3 - 2i) * (3 + 2i)

Z₂ * Z3 / Z₁ = ((-8 * 3 + 8 * 2i) + (8 * 3 + 8 * 2i)) / (3^2 + (-2i)^2) = (16 + 16i) / 13

Therefore, Z₂ * Z3 / Z₁ = (16 + 16i) / 13.

Answer 2

To evaluate Z₂ Z3/Z₁, we need to multiply Z₂ and Z3 and then divide by Z₁.

First, we'll multiply Z₂ and Z3:

Z₂ Z3 = (-2 + 3i)(4 + 2i)
= -8 - 4i + 12i + 6i²
= -8 + 8i + 6(-1)
= -2 + 8i

Next, we'll divide this result by Z₁:

Z₂ Z3 / Z₁ = (-2 + 8i) / (3 - 2i)

We can simplify this fraction using complex numbers:

= ( (-2) * (3 + 2i) + (8i) * (3 - 2i) ) / ( (3 - 2i) * (3 + 2i) )

= (6 + 4i + 16) / (9 + 6)

= 22 / 15

So the answer is Z₂ Z3 / Z₁ = 22/15.