Question
\( \left\lvert\, \leftarrow \quad \begin{array}{l}\text { Let } \mathbf{v}=\langle 9,5\rangle \text {. Carry out the following computation. } \\ \text { Find }|-2 \mathbf{v}|\end{array}\right. \) \( \begin{array}{l}|-2 \mathbf{v}|=\square \\ \text { (Type an exact answer, using radicals as needed.) }\end{array} \)
Ask by Hills Salazar. in the United States
Jan 22,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( |-2 \mathbf{v}| = 2\sqrt{106} \)
Solution
To find the magnitude of \(-2\mathbf{v}\) where \(\mathbf{v} = \langle 9, 5 \rangle\), follow these steps:
1. **Compute \(-2\mathbf{v}\):**
\[
-2\mathbf{v} = -2 \langle 9, 5 \rangle = \langle -18, -10 \rangle
\]
2. **Find the magnitude \(|-2\mathbf{v}|\):**
\[
|-2\mathbf{v}| = \sqrt{(-18)^2 + (-10)^2} = \sqrt{324 + 100} = \sqrt{424}
\]
3. **Simplify \(\sqrt{424}\):**
\[
\sqrt{424} = \sqrt{4 \times 106} = 2\sqrt{106}
\]
So, the exact answer is:
\[
| -2 \mathbf{v} | = 2\sqrt{106}
\]
**Answer:**
\( |-2 \mathbf{v}| = 2\,\sqrt{106} \)
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Simplify this solution Bonus Knowledge
To find \(|-2 \mathbf{v}|\), we first calculate \(-2 \mathbf{v}\). Given \(\mathbf{v} = \langle 9,5\rangle\), \[ -2 \mathbf{v} = -2 \langle 9,5 \rangle = \langle -18, -10 \rangle. \] Next, we calculate the magnitude (or length) of the vector \(\langle -18, -10 \rangle\) using the formula for the magnitude of a vector \(\sqrt{x^2 + y^2}\): \[ |-2 \mathbf{v}| = \sqrt{(-18)^2 + (-10)^2} = \sqrt{324 + 100} = \sqrt{424}. \] We can simplify \(\sqrt{424}\): \[ \sqrt{424} = \sqrt{4 \times 106} = 2\sqrt{106}. \] Putting it all together, we find: \[ |-2 \mathbf{v}| = 2\sqrt{106}. \] So, the final answer is: \[ \boxed{2\sqrt{106}}. \]
