Textbook Question
In Exercises 9–16, letu = 2i - j, v = 3i + j, and w = i + 4j.Find each specified scalar.(4u) ⋅ v
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8. Vectors
Dot Product
3:36 minutesProblem 21
Textbook Question
In Exercises 17–22, find the angle between v and w. Round to the nearest tenth of a degree.v = 6i, w = 5i + 4j
Verified step by step guidance1
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.
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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.
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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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