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

April 10, 2026Mathematics

ScanSolve AI Answer

April 10, 2026

Answer

$\sqrt{317}$

Right, let's go. Step 1: Calculate $3p$ . $3p = 3 \begin{pmatrix} 3 \\ -2 \end{pmatrix} = \begin{pmatrix} 3 \times 3 \\ 3 \times -2 \end{pmatrix} = \begin{pmatrix} 9 \\ -6 \end{pmatrix}$

Step 2: Calculate $2q$ . $2q = 2 \begin{pmatrix} -1 \\ 4 \end{pmatrix} = \begin{pmatrix} 2 \times -1 \\ 2 \times 4 \end{pmatrix} = \begin{pmatrix} -2 \\ 8 \end{pmatrix}$

Step 3: Calculate $3p - 2q$ . $3p - 2q = \begin{pmatrix} 9 \\ -6 \end{pmatrix} - \begin{pmatrix} -2 \\ 8 \end{pmatrix} = \begin{pmatrix} 9 - (-2) \\ -6 - 8 \end{pmatrix} = \begin{pmatrix} 9 + 2 \\ -6 - 8 \end{pmatrix} = \begin{pmatrix} 11 \\ -14 \end{pmatrix}$

Step 4: 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 $\boxed{\sqrt{317}}$ . Send me the next one 📸

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Asked on April 10, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

ScanSolve AI AnswerAnswered on April 10, 2026

Right, let's go. Step 1: Calculate 3p. 3p = 3 3 \\ -2 = 3 × 3 \\ 3 × -2 = 9 \\ -6 Step 2: Calculate 2q. 2q = 2 -1 \\ 4 = 2 × -1 \\ 2 × 4 = -2 \\ 8 Step 3: Calculate 3p - 2q. 3p - 2q = 9 \\ -6 - -2 \\ 8 = 9 - (-2) \\ -6 - 8 = 9 + 2 \\ -6 - 8 = 11 \\ -14 Step 4: Find the magnitude of 3p - 2q. The magnitude of a vector x \\ y is sqrt(x^2 + y^2). |3p - 2q| = | 11 \\ -14 | = sqrt(11^2 + (-14)^2) |3p - 2q| = sqrt(121 + 196) |3p - 2q| = sqrt(317) The final answer is sqrt(317). Send me the next one 📸