If \(p = \begin{pmatrix} 2 \\ -2 \end{pmatrix} \) and \(q = \begin{pmatrix} 3 \\ 4 \end{pmatrix}\), find \(|q - \frac{1}{2}p|\).
A. \(2\sqrt{2}\)
B. \(\sqrt{13}\)
C. \(5\)
D. \(\sqrt{29}\)
Correct Answer: D
Explanation
\(p = \begin{pmatrix} 2 \\ -2 \end{pmatrix} , q = \begin{pmatrix} 3 \\ 4 \end{pmatrix}\)
\(\frac{1}{2}p = \begin{pmatrix} 1 \\ -1 \end{pmatrix}\)
\(q - \frac{1}{2}p = \begin{pmatrix} 3 \\ 4 \end{pmatrix} - \begin{pmatrix} 1 \\ -1 \end{pmatrix} = \begin{pmatrix} 2 \\ 5 \end{pmatrix}\)
\(|q - \frac{1}{2}p| = \sqrt{2^{2} + 5^{2}} = \sqrt{29}\)
