Problem 58 Find the angle to the nearest te... [FREE SOLUTION]

Chapter 7: Problem 58

Find the angle to the nearest tenth of a degree between each given pair of vectors. $$\langle- 2,-5\rangle,\langle 1,-9\rangle$$

Short Answer

Expert verified

Approximately 48.5 degrees

Step by step solution

Find the dot product of the vectors

The dot product of two vectors $\textbf{a} = \langle a_1, a_2\rangle$ and $\textbf{b} = \langle b_1, b_2\rangle$ is calculated using the formula: \[ \textbf{a} \cdot \textbf{b} = a_1b_1 + a_2b_2 \] For vectors $\textbf{a} = \langle-2, -5\rangle$ and $\textbf{b} = \langle 1, -9\rangle$, the dot product is: \[ (-2 \cdot 1) + (-5 \cdot -9) = -2 + 45 = 43 \]

Find the magnitudes of the vectors

The magnitude of a vector $\textbf{a} = \langle a_1, a_2\rangle$ is calculated using the formula: \[ \|\textbf{a}\| = \sqrt{a_1^2 + a_2^2} \] For $\textbf{a} = \langle-2, -5\rangle$: \[ \|\langle-2, -5\rangle\| = \sqrt{(-2)^2 + (-5)^2} = \sqrt{4 + 25} = \sqrt{29} \] For $\textbf{b} = \langle 1, -9\rangle$: \[ \|\langle 1, -9\rangle\| = \sqrt{1^2 + (-9)^2} = \sqrt{1 + 81} = \sqrt{82} \]

Use the dot product and magnitudes to find the cosine of the angle

The cosine of the angle $\theta$ between two vectors is given by: \[ \cos(\theta) = \frac{\textbf{a} \cdot \textbf{b}}{\|\textbf{a}\|\times\| \textbf{b}\|} \] Substituting the values we found: \[ \cos(\theta) = \frac{43}{\sqrt{29} \times \sqrt{82}} = \frac{43}{\sqrt{2378}} \]

Calculate the angle $ \theta $

Use the inverse cosine function to determine the angle $ \theta $: \[ \theta = \cos^{-1}\left(\frac{43}{\sqrt{2378}}\right) \approx 48.5^\circ \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dot Product

The dot product is a key operation in vector mathematics. It's calculated using two vectors: \ \textbf{a} = \langle a_1, a_2\rangle and \textbf{b} = \langle b_1, b_2\rangle. The formula for the dot product is \[ \textbf{a} \cdot \textbf{b} = a_1 b_1 + a_2 b_2 \]. This operation means we multiply corresponding components and then sum the results.\

For example, if we have vectors \textbf{a} = \langle -2, -5 \rangle and \textbf{b} = \langle 1, -9 \rangle, the dot product is:

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91影视

Find the angle to the nearest tenth of a degree between each given pair of vectors. $$\langle- 2,-5\rangle,\langle 1,-9\rangle$$

Short Answer

Step by step solution

Find the dot product of the vectors

Find the magnitudes of the vectors

Use the dot product and magnitudes to find the cosine of the angle

Calculate the angle \( \theta \)

Key Concepts

Dot Product

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