Textbook Question
Find the angle between each pair of vectors. Round to two decimal places as necessary.
3i + 4j, j
Ch. 7 - Applications of Trigonometry and Vectors
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 8, Problem 61
Find the angle between each pair of vectors. Round to two decimal places as necessary.
〈1, 6〉, 〈-1, 7〉
Verified step by step guidance1
Identify the two vectors given: \( \mathbf{u} = \langle 1, 6 \rangle \) and \( \mathbf{v} = \langle -1, 7 \rangle \).
Recall the formula for the angle \( \theta \) between two vectors \( \mathbf{u} \) and \( \mathbf{v} \):
\[\cos(\theta) = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|}\]
Calculate the dot product \( \mathbf{u} \cdot \mathbf{v} \) using the formula:
\[\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2\]
where \( u_1, u_2 \) and \( v_1, v_2 \) are the components of \( \mathbf{u} \) and \( \mathbf{v} \) respectively.
Find the magnitudes of each vector using:
\[\|\mathbf{u}\| = \sqrt{u_1^2 + u_2^2} \quad \text{and} \quad \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2}\]
Substitute the dot product and magnitudes into the cosine formula, then use the inverse cosine function to find the angle \( \theta = \cos^{-1} \left( \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|} \right) \). Finally, round your answer to two decimal places.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Dot Product of Vectors
The dot product is a scalar value obtained by multiplying corresponding components of two vectors and summing the results. For vectors 〈x1, y1〉 and 〈x2, y2〉, it is calculated as x1*x2 + y1*y2. This product is essential for finding the angle between vectors.
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Introduction to Dot Product
Magnitude of a Vector
The magnitude (or length) of a vector 〈x, y〉 is found using the formula √(x² + y²). It represents the distance from the origin to the point defined by the vector and is used to normalize vectors when calculating angles.
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04:44Finding Magnitude of a Vector
Angle Between Two Vectors
The angle θ between two vectors can be found using the formula cos(θ) = (dot product) / (product of magnitudes). By taking the inverse cosine (arccos) of this ratio, we obtain the angle in radians or degrees, which can then be rounded as required.
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04:33Find the Angle Between Vectors
Related Practice
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