Which of the following is equivalent to $-2 i(6-7 i)$ ?
To find which expression is equivalent to \(-2i(6 - 7i)\), we first represent the complex number \(-2i\) in the standard form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. A complex number \(z = x + iy\) has a real part \(x\) and an imaginary part \(y\). For the term \(-2i\), the real part is \(0\) and the imaginary part is \(-2\). Thus, we can write \(-2i = 0 - 2i\).
Now that we have \(-2i = 0 - 2i\), we can substitute this into the original expression. The original expression is \(-2i(6 - 7i)\). Substituting the standard form, we get \((0 - 2i)(6 - 7i)\). We compare this result with the provided options:
Option c matches our derived expression \((0 - 2i)(6 - 7i)\) perfectly.
To be absolutely sure, we can expand the original expression and the identified option.
Expanding \(-2i(6 - 7i)\):
\[-2i \times 6 - (-2i) \times 7i = -12i + 14i^2\]
Since \(i^2 = -1\), this becomes:
\[-12i + 14(-1) = -14 - 12i\]
Now expanding option c, \((0 - 2i)(6 - 7i)\):
\[0 \times (6 - 7i) - 2i(6 - 7i) = 0 - 12i + 14i^2 = -12i - 14 = -14 - 12i\]
Both expressions are equivalent to \(-14 - 12i\). Therefore, option c is the correct choice.
Complex number operations are fundamental in fields like electrical engineering, where they are used to analyze AC circuits. For instance, the voltage across an inductor is proportional to the current multiplied by \(i \times \text{angular frequency} \times \text{inductance}\). Identifying equivalent forms of these complex products helps simplify complex circuit models, allowing engineers to calculate impedance efficiently without unnecessary clutter in their equations.