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Ch. 7 - Applications of Trigonometry and Vectors
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 7 - Applications of Trigonometry and VectorsProblema 61
Capitolo 8, Problema 61

Find the angle between each pair of vectors. Round to two decimal places as necessary.
〈1, 6〉, 〈-1, 7〉

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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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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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05:40
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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04:44
Finding 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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04:33
Find the Angle Between Vectors
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