Textbook Question
In Exercises 40–41, use the dot product to determine whether v and w are orthogonal.
v = 12i - 8j, w = 2i + 3j
Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 4, Problem 38
In Exercises 37–39, find the dot product v ⋅ w. Then find the angle between v and w to the nearest tenth of a degree.
v = 2i + 4j, w = 6i - 11j
Verified step by step guidance1
Identify the components of the vectors \( \mathbf{v} = 2\mathbf{i} + 4\mathbf{j} \) and \( \mathbf{w} = 6\mathbf{i} - 11\mathbf{j} \). Here, \( \mathbf{v} = (2, 4) \) and \( \mathbf{w} = (6, -11) \).
Calculate the dot product \( \mathbf{v} \cdot \mathbf{w} \) using the formula \( \mathbf{v} \cdot \mathbf{w} = v_1 w_1 + v_2 w_2 \). Substitute the components to get \( 2 \times 6 + 4 \times (-11) \).
Find the magnitudes of \( \mathbf{v} \) and \( \mathbf{w} \) using the formula \( \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2} \) and similarly for \( \mathbf{w} \). Calculate \( \|\mathbf{v}\| = \sqrt{2^2 + 4^2} \) and \( \|\mathbf{w}\| = \sqrt{6^2 + (-11)^2} \).
Use the dot product and magnitudes to find the cosine of the angle \( \theta \) between the vectors with the formula \( \cos(\theta) = \frac{\mathbf{v} \cdot \mathbf{w}}{\|\mathbf{v}\| \|\mathbf{w}\|} \).
Finally, find the angle \( \theta \) by taking the inverse cosine (arccos) of the value found in the previous step, and convert the result to degrees if necessary, rounding to the nearest tenth of a degree.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Dot Product of Two Vectors
The dot product is an algebraic operation that takes two vectors and returns a scalar. It is calculated by multiplying corresponding components and summing the results, e.g., for vectors v = (v1, v2) and w = (w1, w2), v ⋅ w = v1w1 + v2w2. This operation measures how much one vector extends in the direction of another.
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Magnitude of a Vector
The magnitude (or length) of a vector is the distance from the origin to the point represented by the vector. For a vector v = (v1, v2), its magnitude is |v| = √(v1² + v2²). Magnitudes are essential for normalizing vectors and calculating angles between them.
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Angle Between Two Vectors
The angle θ between two vectors v and w can be found using the dot product formula: v ⋅ w = |v||w|cos(θ). Rearranging gives θ = cos⁻¹((v ⋅ w) / (|v||w|)). This formula relates the dot product and magnitudes to the cosine of the angle, allowing calculation of the angle in degrees or radians.
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Related Practice
Textbook Question
In Exercises 33–38, find the area of the triangle having the given measurements. Round to the nearest square unit. C = 102°, a = 16 meters, b = 20 meters
Textbook Question
In Exercises 33–38, find projᵥᵥ v. Then decompose v into two vectors, v₁ and v₂, where v₁ is parallel to w and v₂ is orthogonal to w.
v = i + 2j, w = 3i + 6j
Textbook Question
In Exercises 39–42, let u = -i + j, v = 3i - 2j, and w = -5j. Find each specified scalar or vector.
5u ⋅ (3v - 4w)
Textbook Question
In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.
||w - u||
Textbook Question
In Exercises 39–46, find the unit vector that has the same direction as the vector v.
v = 6i