Question

1. given vector write its direction and location
2. does (a + bi)(a - bi) always equal a² + b²? why?

look back at #1.

Explanation:

Step1: Expand the complex - number product

Use the FOIL method. \((a + bi)(a - bi)=a\times a+a\times(-bi)+bi\times a+bi\times(-bi)\).
\[

$$\begin{align*} &=a^{2}-abi + abi - b^{2}i^{2}\\ \end{align*}$$

\]

Step2: Simplify the expression

Since \(i^{2}=- 1\), substitute \(i^{2}\) into the above - expression.
\[

$$\begin{align*} &=a^{2}-b^{2}\times(-1)\\ &=a^{2}+b^{2} \end{align*}$$

\]

Answer:

Yes, \((a + bi)(a - bi)\) always equals \(a^{2}+b^{2}\) because when we expand \((a + bi)(a - bi)\) using the FOIL method and simplify by substituting \(i^{2}=-1\), we get \(a^{2}+b^{2}\).