Given vectors a = 2 4 and b = -3 4 , find the magnitude of vector a - 2b.

May 17, 2026Mathematics

ScanSolve AI Answer

May 17, 2026

Answer

$\mathbf{4\sqrt{5}}$

Step 1: Calculate $2b$ . $2b = 2 \begin{pmatrix} -3 \\ 4 \end{pmatrix} = \begin{pmatrix} 2 \times -3 \\ 2 \times 4 \end{pmatrix} = \begin{pmatrix} -6 \\ 8 \end{pmatrix}$

Step 2: Calculate $a - 2b$ . $a - 2b = \begin{pmatrix} 2 \\ 4 \end{pmatrix} - \begin{pmatrix} -6 \\ 8 \end{pmatrix} = \begin{pmatrix} 2 - (-6) \\ 4 - 8 \end{pmatrix} = \begin{pmatrix} 2 + 6 \\ -4 \end{pmatrix} = \begin{pmatrix} 8 \\ -4 \end{pmatrix}$

Step 3: Calculate the magnitude $|a - 2b|$ . The magnitude of a vector $\begin{pmatrix} x \\ y \end{pmatrix}$ is $\sqrt{x^2 + y^2}$ . $|a - 2b| = \left| \begin{pmatrix} 8 \\ -4 \end{pmatrix} \right| = \sqrt{8^2 + (-4)^2}$ $|a - 2b| = \sqrt{64 + 16} = \sqrt{80}$ $|a - 2b| = \sqrt{16 \times 5} = 4\sqrt{5}$

The value of $|a - 2b|$ is $\boxed{4\sqrt{5}}$ .

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Calculate 2b. 2b = 2 -3 \\ 4 = 2 × -3 \\ 2 × 4 = -6 \\ 8 Step 2: Calculate a - 2b.

Asked on May 17, 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 May 17, 2026

Step 1: Calculate 2b. 2b = 2 -3 \\ 4 = 2 × -3 \\ 2 × 4 = -6 \\ 8 Step 2: Calculate a - 2b. a - 2b = 2 \\ 4 - -6 \\ 8 = 2 - (-6) \\ 4 - 8 = 2 + 6 \\ -4 = 8 \\ -4 Step 3: Calculate the magnitude |a - 2b|. The magnitude of a vector x \\ y is sqrt(x^2 + y^2). |a - 2b| = | 8 \\ -4 | = sqrt(8^2 + (-4)^2) |a - 2b| = sqrt(64 + 16) = sqrt(80) |a - 2b| = sqrt(16 × 5) = 4sqrt(5) The value of |a - 2b| is 4sqrt(5). Send me the next one 📸