Cole Wilson
07/14/2024 · Primary School
3.3 Find the cosine of the angle between the vectors \( \mathbf{C}=(2,-3) \) and \( \mathbf{D}=(3,1) \)
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Step-by-step Solution
To find the cosine of the angle between the vectors \(\mathbf{C} = (2, -3)\) and \(\mathbf{D} = (3, 1)\), we can use the formula:
\[
\cos \theta = \frac{\mathbf{C} \cdot \mathbf{D}}{\|\mathbf{C}\| \|\mathbf{D}\|}
\]
where \(\mathbf{C} \cdot \mathbf{D}\) is the dot product of the vectors, and \(\|\mathbf{C}\|\) and \(\|\mathbf{D}\|\) are the magnitudes (norms) of the vectors.
1. **Calculate the dot product \(\mathbf{C} \cdot \mathbf{D}\)**:
\[
\mathbf{C} \cdot \mathbf{D} = (2)(3) + (-3)(1) = 6 - 3 = 3
\]
2. **Calculate the magnitudes \(\|\mathbf{C}\|\) and \(\|\mathbf{D}\|\)**:
\[
\|\mathbf{C}\| = \sqrt{(2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13}
\]
\[
\|\mathbf{D}\| = \sqrt{(3)^2 + (1)^2} = \sqrt{9 + 1} = \sqrt{10}
\]
3. **Substitute the values into the cosine formula**:
\[
\cos \theta = \frac{3}{\sqrt{13} \cdot \sqrt{10}} = \frac{3}{\sqrt{130}}
\]
Thus, the cosine of the angle between the vectors \(\mathbf{C}\) and \(\mathbf{D}\) is:
\[
\cos \theta = \frac{3}{\sqrt{130}}
\]
Quick Answer
\(\cos \theta = \frac{3}{\sqrt{130}}\)
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