Divide.
$\frac{-3 i}{4+5 i}$
Write your answer as a complex number in standard form.
The objective is to divide the complex number \(-3i\) by \(4+5i\) and express the result in the standard form \(a+bi\), where \(a\) and \(b\) are real numbers. We are given the expression:
\[\frac{-3i}{4+5i}\]
To simplify a quotient of complex numbers, we need to eliminate the imaginary part from the denominator. This is achieved by multiplying both the numerator and the denominator by the complex conjugate of the denominator. For a complex number \(z = x+yi\), its conjugate is \(\bar{z} = x-yi\). Thus, the conjugate of \(4+5i\) is \(4-5i\).
We multiply the expression by the conjugate of the denominator divided by itself (which is equivalent to multiplying by 1):
\[\frac{-3i}{4+5i} \times \frac{4-5i}{4-5i} = \frac{-3i(4-5i)}{(4+5i)(4-5i)}\]
First, let's simplify the denominator using the property that for any complex number \(z\), \(z \cdot \bar{z} = x^2+y^2\):
\[(4+5i)(4-5i) = 4^2 + 5^2 = 16 + 25 = 41\]
Next, let's simplify the numerator by distributing \(-3i\):
\[-3i(4-5i) = -3i(4) - 3i(-5i) = -12i + 15i^2\]
Since \(i^2 = -1\), the numerator becomes:
\[-12i + 15(-1) = -12i - 15 = -15 - 12i\]
Combining the simplified numerator and denominator, we have:
\[\frac{-15 - 12i}{41}\]
To express this in the standard form \(a+bi\), we divide both the real and imaginary parts of the numerator by the denominator:
\[-\frac{15}{41} - \frac{12}{41}i\]
Thus, the final result in standard form is \(\boxed{-\frac{15}{41} - \frac{12}{41}i}\).
In electrical engineering, complex numbers are used to represent impedance in AC circuits. Dividing complex numbers is a common operation when calculating the total impedance of components arranged in parallel or when analyzing the relationship between voltage and current. By converting quotients into the standard form \(a+bi\), engineers can easily separate the real part (resistance) from the imaginary part (reactance), allowing for more precise design and troubleshooting of electronic circuits.