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Given vectors p = [3, -2] and q = [-1, 4], calculate the magnitude of 3p - 2q.

April 10, 2026Mathematics

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Answer

\sqrt{317}

4 steps

Calculate $3p$.
$3p = 3 \begin{pmatrix} 3 \\ -2 \end{pmatrix} = \begin{pmatrix} 3 \times 3 \\ 3 \times -2 \end{pmatrix} = \begin{pmatrix} 9 \\ -6 \end{pmatrix}$
Calculate $2q$.
$2q = 2 \begin{pmatrix} -1 \\ 4 \end{pmatrix} = \begin{pmatrix} 2 \times -1 \\ 2 \times 4 \end{pmatrix} = \begin{pmatrix} -2 \\ 8 \end{pmatrix}$
Calculate $3p - 2q$ .
$3p - 2q = \begin{pmatrix} 9 \\ -6 \end{pmatrix} - \begin{pmatrix} -2 \\ 8 \end{pmatrix} = \begin{pmatrix} 9 - (-2) \\ -6 - 8 \end{pmatrix} = \begin{pmatrix} 9 + 2 \\ -6 - 8 \end{pmatrix} = \begin{pmatrix} 11 \\ -14 \end{pmatrix}$
Find the magnitude of $3p - 2q$ .
The magnitude of a vector $\begin{pmatrix} x \\ y \end{pmatrix}$ is $\sqrt{x^2 + y^2}$ .
$|3p - 2q| = \left| \begin{pmatrix} 11 \\ -14 \end{pmatrix} \right| = \sqrt{11^2 + (-14)^2}$ $|3p - 2q| = \sqrt{121 + 196}$ $|3p - 2q| = \sqrt{317}$
The final answer is $\sqrt{317}$ .