Textbook Question
In Exercises 5–8, let v = -5i + 2j and w = 2i - 4j Find the specified vector, scalar, or angle. 3v - 4w
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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 6
In Exercises 5–8, let v = -5i + 2j and w = 2i - 4j Find the specified vector, scalar, or angle. v ⋅ w
Verified step by step guidance1
Recall that the dot product of two vectors \( \mathbf{v} = v_1 \mathbf{i} + v_2 \mathbf{j} \) and \( \mathbf{w} = w_1 \mathbf{i} + w_2 \mathbf{j} \) is given by the formula:
\[ \mathbf{v} \cdot \mathbf{w} = v_1 w_1 + v_2 w_2 \]
Identify the components of the given vectors:
For \( \mathbf{v} = -5 \mathbf{i} + 2 \mathbf{j} \), we have \( v_1 = -5 \) and \( v_2 = 2 \).
For \( \mathbf{w} = 2 \mathbf{i} - 4 \mathbf{j} \), we have \( w_1 = 2 \) and \( w_2 = -4 \).
Substitute the components into the dot product formula:
\[ \mathbf{v} \cdot \mathbf{w} = (-5)(2) + (2)(-4) \]
Simplify the expression by performing the multiplications and then adding the results:
\[ (-5)(2) + (2)(-4) = -10 + (-8) \]
Add the two products to find the dot product value:
\[ -10 + (-8) = -18 \]
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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 and summing the results, e.g., for vectors v = (v1, v2) and w = (w1, w2), v ⋅ w = v1w1 + v2w2.
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Introduction to Dot Product
Vector Components and Notation
Vectors in two dimensions are expressed in terms of unit vectors i and j, representing the x and y directions respectively. Understanding how to identify and manipulate these components is essential for performing operations like the dot product.
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Geometric Interpretation of the Dot Product
The dot product relates to the angle between two vectors: v ⋅ w = |v||w|cosθ, where θ is the angle between v and w. This relationship helps in finding the angle or understanding the projection of one vector onto another.
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Related Practice
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. C = 50°, a = 3, c = 1
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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.
Textbook Question
In Exercises 5–8, each expression is the right side of the formula for cos (α - β) with particular values for α and β. c. Find the exact value of the expression.cos 50° cos 20° + sin 50° sin 20°
Textbook Question
In Exercises 1–8, solve each triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree.
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Textbook Question
In Exercises 5–12, sketch each vector as a position vector and find its magnitude. v = i - j