If \(s = 3i - j\) and \(t = 2i + 3j\), find \((t - 3s).(t + 3s)\).

If \(s = 3i - j\) and \(t = 2i + 3j\), find \((t - 3s).(t + 3s)\).

Answer Details

First, let's find the value of the expressions inside the parentheses: \begin{align*} t - 3s &= 2i + 3j - 3(3i - j)\\ &= 2i + 3j - 9i + 3j\\ &= -7i + 6j\\ \\ t + 3s &= 2i + 3j + 3(3i - j)\\ &= 2i + 3j + 9i - 3j\\ &= 11i \end{align*} Now we can substitute these expressions into the dot product: \begin{align*} (t - 3s).(t + 3s) &= (-7i + 6j).(11i)\\ &= -77i + 66j\\ \end{align*} Therefore, the answer is -77.