Textbook Question
In Exercises 1–4, u and v have the same direction. In each exercise: Find ||u||.
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Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 4, Problem 3
In Exercises 1–8, use the given vectors to find v⋅w and v⋅v. v = 5i - 4j, w = -2i - j
Verified step by step guidance1
Identify the components of the vectors \( \mathbf{v} = 5\mathbf{i} - 4\mathbf{j} \) and \( \mathbf{w} = -2\mathbf{i} - \mathbf{j} \). Here, \( \mathbf{v} = (5, -4) \) and \( \mathbf{w} = (-2, -1) \).
Recall the formula for the dot product of two vectors \( \mathbf{a} = (a_1, a_2) \) and \( \mathbf{b} = (b_1, b_2) \):
\[
\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2
\]
Calculate \( \mathbf{v} \cdot \mathbf{w} \) by multiplying corresponding components and adding the results:
\[
\mathbf{v} \cdot \mathbf{w} = (5)(-2) + (-4)(-1)
\]
Recall that \( \mathbf{v} \cdot \mathbf{v} \) is the dot product of \( \mathbf{v} \) with itself, which gives the square of its magnitude:
\[
\mathbf{v} \cdot \mathbf{v} = 5^2 + (-4)^2
\]
Perform the arithmetic operations in the expressions from steps 3 and 4 to find the values of \( \mathbf{v} \cdot \mathbf{w} \) and \( \mathbf{v} \cdot \mathbf{v} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Dot Product of Vectors
The dot product is an algebraic operation that takes two vectors and returns a scalar. It is calculated by multiplying corresponding components of the vectors and summing the results. For vectors v = ai + bj and w = ci + dj, the dot product is v·w = ac + bd.
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Introduction to Dot Product
Vector Components
Vectors in two dimensions can be expressed in terms of their components along the i (x-axis) and j (y-axis) unit vectors. Understanding how to identify and use these components is essential for operations like addition, subtraction, and dot product.
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03:55Position Vectors & Component Form
Dot Product of a Vector with Itself
The dot product of a vector with itself, v·v, gives the square of its magnitude. It is calculated by summing the squares of its components, i.e., v·v = a² + b² for v = ai + bj. This is useful for finding the length or magnitude of the vector.
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05:40Introduction to Dot Product
Related Practice
Textbook Question
Be sure that you've familiarized yourself with the first set of formulas presented in this section by working C1–C4 in the Concept and Vocabulary Check. In Exercises 1–8, use the appropriate formula to express each product as a sum or difference.cos 7x cos 3x
Textbook Question
In Exercises 1–12, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. If no triangle exists, state 'no triangle.' If two triangles exist, solve each triangle. B = 66°, a = 17, c = 12
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Textbook Question
In Exercises 1–4, u and v have the same direction. In each exercise: Find ||v||.
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Textbook Question
In Exercises 1–8, solve each triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree.
1
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Textbook Question
In Exercises 1–12, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. If no triangle exists, state 'no triangle.' If two triangles exist, solve each triangle. B = 107°, C = 30°, c = 126
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