In Exercises 17–22, find the angle between v and w. Round to the nearest tenth of a degree.v = -3i + 2j, w = 4i - j

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Step 1: Recall the formula for the angle \( \theta \) between two vectors \( \mathbf{v} \) and \( \mathbf{w} \): \( \cos \theta = \frac{\mathbf{v} \cdot \mathbf{w}}{\|\mathbf{v}\| \|\mathbf{w}\|} \).

Step 2: Calculate the dot product \( \mathbf{v} \cdot \mathbf{w} \). For vectors \( \mathbf{v} = -3i + 2j \) and \( \mathbf{w} = 4i - j \), the dot product is \( (-3)(4) + (2)(-1) \).

Step 3: Find the magnitudes of \( \mathbf{v} \) and \( \mathbf{w} \). The magnitude of \( \mathbf{v} \) is \( \sqrt{(-3)^2 + 2^2} \) and the magnitude of \( \mathbf{w} \) is \( \sqrt{4^2 + (-1)^2} \).

Step 4: Substitute the dot product and magnitudes into the formula: \( \cos \theta = \frac{\text{dot product}}{\text{magnitude of } \mathbf{v} \times \text{magnitude of } \mathbf{w}} \).

Step 5: Use the inverse cosine function to find \( \theta \): \( \theta = \cos^{-1}(\text{calculated value}) \). Round the result to the nearest tenth of a degree.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Dot Product

The dot product of two vectors is a scalar value that is calculated by multiplying their corresponding components and summing the results. It is given by the formula v · w = v_x * w_x + v_y * w_y for 2D vectors. The dot product is crucial for finding the angle between two vectors, as it relates to the cosine of the angle through the equation v · w = |v| |w| cos(θ).

Magnitude of a Vector

The magnitude of a vector is a measure of its length and is calculated using the formula |v| = √(v_x² + v_y²) for 2D vectors. Understanding how to compute the magnitude is essential for determining the angle between vectors, as it is used in the dot product formula. The magnitudes of both vectors must be known to find the cosine of the angle between them.

Cosine of the Angle

The cosine of the angle between two vectors can be derived from the dot product and the magnitudes of the vectors. Specifically, cos(θ) = (v · w) / (|v| |w|). This relationship allows us to find the angle θ by taking the inverse cosine (arccos) of the calculated cosine value. This concept is fundamental in trigonometry and vector analysis.

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