Carlson Marsh
12/13/2023 · High School
Given that \( \mathbf{b}=\binom{-7}{8}, \mathbf{c}=\binom{9}{-4} \) anc \( \mathbf{a}=3 \mathbf{c}-9 \mathbf{b} \), find the magnitude of \( \mathbf{a} \) correct to 2 decimal places.
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Tutor-Verified Answer
Step-by-step Solution
To find the vector \(\mathbf{a}\), we start by calculating \(3 \mathbf{c}\) and \(-9 \mathbf{b}\).
First, we calculate \(3 \mathbf{c}\):
\[
\mathbf{c} = \binom{9}{-4}
\]
\[
3 \mathbf{c} = 3 \binom{9}{-4} = \binom{3 \cdot 9}{3 \cdot -4} = \binom{27}{-12}
\]
Next, we calculate \(-9 \mathbf{b}\):
\[
\mathbf{b} = \binom{-7}{8}
\]
\[
-9 \mathbf{b} = -9 \binom{-7}{8} = \binom{-9 \cdot -7}{-9 \cdot 8} = \binom{63}{-72}
\]
Now we can find \(\mathbf{a}\):
\[
\mathbf{a} = 3 \mathbf{c} - 9 \mathbf{b} = \binom{27}{-12} + \binom{63}{-72}
\]
To add these vectors, we add their corresponding components:
\[
\mathbf{a} = \binom{27 + 63}{-12 - 72} = \binom{90}{-84}
\]
Next, we find the magnitude of \(\mathbf{a}\):
\[
\|\mathbf{a}\| = \sqrt{90^2 + (-84)^2}
\]
Calculating \(90^2\) and \((-84)^2\):
\[
90^2 = 8100
\]
\[
(-84)^2 = 7056
\]
Now, we add these two results:
\[
\|\mathbf{a}\| = \sqrt{8100 + 7056} = \sqrt{15156}
\]
Now we compute \(\sqrt{15156}\):
\[
\sqrt{15156} \approx 123.16
\]
Thus, the magnitude of \(\mathbf{a}\) correct to 2 decimal places is:
\[
\boxed{123.16}
\]
Quick Answer
The magnitude of \(\mathbf{a}\) is 123.16.
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