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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 63
Capitolo 8, Problema 63

Find the angle between each pair of vectors. Round to two decimal places as necessary.
3i + 4j, j

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Identify the two vectors given: the first vector is \(\mathbf{v_1} = 3\mathbf{i} + 4\mathbf{j}\) and the second vector is \(\mathbf{v_2} = \mathbf{j}\), which can be written as \(0\mathbf{i} + 1\mathbf{j}\).
Recall the formula for the angle \(\theta\) between two vectors \(\mathbf{a}\) and \(\mathbf{b}\): \(\cos(\theta) = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \|\mathbf{b}\|}\), where \(\mathbf{a} \cdot \mathbf{b}\) is the dot product and \(\|\mathbf{a}\|\), \(\|\mathbf{b}\|\) are the magnitudes of the vectors.
Calculate the dot product of \(\mathbf{v_1}\) and \(\mathbf{v_2}\): \(\mathbf{v_1} \cdot \mathbf{v_2} = (3)(0) + (4)(1) = 4\).
Find the magnitudes of each vector: \(\|\mathbf{v_1}\| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16}\), \(\|\mathbf{v_2}\| = \sqrt{0^2 + 1^2} = 1\).
Substitute the dot product and magnitudes into the cosine formula and solve for \(\theta\): \(\cos(\theta) = \frac{4}{\|\mathbf{v_1}\| \times 1}\), then find \(\theta\) by taking the inverse cosine (arccos) of the result. Finally, round your answer to two decimal places.

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Vector Components and Representation

Vectors in two dimensions can be expressed using unit vectors i and j, representing the x and y components respectively. For example, the vector 3i + 4j has components (3, 4), which helps in calculating magnitude and direction.
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Position Vectors & Component Form

Dot Product of Vectors

The dot product of two vectors is a scalar value found by multiplying corresponding components and summing the results. It is used to determine the angle between vectors through the formula: dot product = |A||B|cos(θ).
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Introduction to Dot Product

Calculating the Angle Between Vectors

The angle θ between two vectors can be found using the dot product formula: θ = arccos[(A·B) / (|A||B|)]. This requires computing the dot product and magnitudes of both vectors, then applying the inverse cosine function.
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Find the Angle Between Vectors
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