Textbook Question
In Exercises 45–50, determine whether v and w are parallel, orthogonal, or neither. v = 3i - 5j, w = 6i - 10j
Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 4, Problem 43
In Exercises 43–44, find the angle, in degrees, between v and w.
v = 2 cos(4π/3) i + 2 sin(4π/3) j, w = 3 cos(3π/2) i + 3 sin(3π/2) j
Verified step by step guidance1
Identify the vectors \( \mathbf{v} = 2 \cos(4\pi) \mathbf{i} + 2 \sin(4\pi) \mathbf{j} \) and \( \mathbf{w} = 3 \cos(3\pi) \mathbf{i} + 3 \sin(3\pi) \mathbf{j} \). These are given in component form using trigonometric functions.
Recall that the angle \( \theta \) between two vectors \( \mathbf{v} \) and \( \mathbf{w} \) can be found using the dot product formula:
\[ \mathbf{v} \cdot \mathbf{w} = \|\mathbf{v}\| \|\mathbf{w}\| \cos(\theta) \]
which rearranges to
\[ \theta = \cos^{-1} \left( \frac{\mathbf{v} \cdot \mathbf{w}}{\|\mathbf{v}\| \|\mathbf{w}\|} \right) \].
Calculate the components of \( \mathbf{v} \) and \( \mathbf{w} \) by evaluating the cosine and sine values at the given angles:
- \( \cos(4\pi) \) and \( \sin(4\pi) \)
- \( \cos(3\pi) \) and \( \sin(3\pi) \).
Compute the dot product \( \mathbf{v} \cdot \mathbf{w} \) using the formula:
\[ \mathbf{v} \cdot \mathbf{w} = v_x w_x + v_y w_y \]
where \( v_x, v_y \) and \( w_x, w_y \) are the components of \( \mathbf{v} \) and \( \mathbf{w} \) respectively.
Find the magnitudes \( \|\mathbf{v}\| \) and \( \|\mathbf{w}\| \) using:
\[ \|\mathbf{v}\| = \sqrt{v_x^2 + v_y^2} \quad \text{and} \quad \|\mathbf{w}\| = \sqrt{w_x^2 + w_y^2} \]
Finally, substitute these values into the formula for \( \theta \) and solve for the angle in degrees.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Representation in the Plane
Vectors in the plane can be expressed using their components along the i (x-axis) and j (y-axis) unit vectors. Here, each vector is given in terms of cosine and sine functions, which correspond to the x and y components based on an angle from the positive x-axis. Understanding this allows you to interpret the vectors geometrically and perform calculations.
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Introduction to Vectors
Dot Product and Angle Between Vectors
The dot product of two vectors relates their magnitudes and the cosine of the angle between them: v · w = |v||w|cos(θ). By computing the dot product and magnitudes, you can solve for the angle θ between vectors. This is essential for finding the angle in degrees between v and w.
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04:33Find the Angle Between Vectors
Trigonometric Functions and Angle Measures
Cosine and sine functions describe the coordinates of points on the unit circle at given angles, often in radians. Converting these angles and understanding their periodicity helps simplify vector components and calculate exact values. Recognizing angles like 3π and 4π radians is key to evaluating the vector components correctly.
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6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
In Exercises 39–46, find the unit vector that has the same direction as the vector v.
v = 4i - 2j
Textbook Question
In Exercises 39–46, find the unit vector that has the same direction as the vector v.
v = 3i - 2j
Textbook Question
In Exercises 42–43, find projᵥᵥv. Then decompose v into two vectors, v₁ and v₂ where v₁ is parallel to w and v₂ is orthogonal to w.
v = -2i + 5j, w = 5i + 4j
Textbook Question
In Exercises 39–46, find the unit vector that has the same direction as the vector v.
v = 8i - 6j
Textbook Question
In Exercises 43–44, use the given measurements to solve the following triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree. a = 400, b = 300