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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 55
Capitolo 8, Problema 55

Find the angle between each pair of vectors. Round to two decimal places as necessary.
〈2, 1〉, 〈-3, 1〉

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1
Identify the two vectors given: \( \mathbf{u} = \langle 2, 1 \rangle \) and \( \mathbf{v} = \langle -3, 1 \rangle \).
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: \[\|\mathbf{u}\| = \sqrt{u_1^2 + u_2^2}\] and \[\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2}\]
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 〈a, b〉 and 〈c, d〉, the dot product is ac + bd. It is essential for finding the angle between vectors.
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Percorso guidato
05:40
Introduction to Dot Product

Magnitude of a Vector

The magnitude (or length) of a vector 〈x, y〉 is calculated 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 finding angles.
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Percorso guidato
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 value, we get the angle in radians or degrees, which can then be rounded as needed.
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Percorso guidato
04:33
Find the Angle Between Vectors
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