In Exercises 17–22, find the angle between v and w. Round to the nearest tenth of a degree.v = 6i, w = 5i + 4j

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insert step 1: Understand that the angle \( \theta \) between two vectors \( \mathbf{v} \) and \( \mathbf{w} \) can be found using the dot product formula: \( \cos \theta = \frac{\mathbf{v} \cdot \mathbf{w}}{\|\mathbf{v}\| \|\mathbf{w}\|} \).

insert step 2: Calculate the dot product \( \mathbf{v} \cdot \mathbf{w} \). For vectors \( \mathbf{v} = 6\mathbf{i} \) and \( \mathbf{w} = 5\mathbf{i} + 4\mathbf{j} \), the dot product is \( 6 \times 5 + 0 \times 4 = 30 \).

insert step 3: Find the magnitudes of \( \mathbf{v} \) and \( \mathbf{w} \). The magnitude of \( \mathbf{v} = 6\mathbf{i} \) is \( \|\mathbf{v}\| = \sqrt{6^2} = 6 \). The magnitude of \( \mathbf{w} = 5\mathbf{i} + 4\mathbf{j} \) is \( \|\mathbf{w}\| = \sqrt{5^2 + 4^2} = \sqrt{41} \).

insert step 4: Substitute the values into the cosine formula: \( \cos \theta = \frac{30}{6 \times \sqrt{41}} \).

insert step 5: Use the inverse cosine function to find \( \theta \): \( \theta = \cos^{-1}\left(\frac{30}{6 \times \sqrt{41}}\right) \). Round \( \theta \) 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 square root of the sum of the squares of its components. For example, the magnitude of vector v = 6i is |v| = 6, while for w = 5i + 4j, it is |w| = √(5² + 4²). Knowing the magnitudes of both vectors is crucial for applying the cosine formula to find the angle between them.

Cosine of the Angle Between Vectors

The cosine of the angle θ 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 calculate the angle by taking the inverse cosine (arccos) of the resulting value, which is necessary for solving the given problem.

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