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

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insert step 1: Start by recalling the formula for the angle \( \theta \) between two vectors \( \mathbf{v} \) and \( \mathbf{w} \): \( \cos \theta = \frac{\mathbf{v} \cdot \mathbf{w}}{\|\mathbf{v}\| \|\mathbf{w}\|} \).

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

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

insert step 4: Substitute the dot product and magnitudes into the formula \( \cos \theta = \frac{\mathbf{v} \cdot \mathbf{w}}{\|\mathbf{v}\| \|\mathbf{w}\|} \) to find \( \cos \theta \).

insert step 5: Use the inverse cosine function to find \( \theta \), the angle between the vectors, and round 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 is a mathematical operation that takes two vectors and returns a scalar. It is calculated as the sum of the products of their corresponding components. For vectors v and w, the dot product can be used to find the cosine of the angle between them, which is essential for determining the angle itself.

Magnitude of a Vector

The magnitude of a vector is a measure of its length and is calculated using the formula √(x² + y²) for a 2D vector with components x and y. Knowing the magnitudes of both vectors is crucial for applying the cosine formula to find the angle between them, as it normalizes the dot product result.

Cosine of the Angle

The cosine of the angle between two vectors can be found using the formula cos(θ) = (v · w) / (|v| |w|), where v · w is the dot product and |v| and |w| are the magnitudes of the vectors. This relationship allows us to derive the angle θ by taking the inverse cosine (arccos) of the calculated value, which is necessary for solving the problem.

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