Textbook Question
In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.
-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 29
In Exercises 23–32, use the dot product to determine whether v and w are orthogonal.
v = 3i, w = -4i
Verified step by step guidance1
Recall that two vectors \( \mathbf{v} \) and \( \mathbf{w} \) are orthogonal if their dot product is zero, i.e., \( \mathbf{v} \cdot \mathbf{w} = 0 \).
Write the given vectors in component form: \( \mathbf{v} = 3\mathbf{i} = (3, 0) \) and \( \mathbf{w} = -4\mathbf{i} = (-4, 0) \).
Calculate the dot product using the formula \( \mathbf{v} \cdot \mathbf{w} = v_x w_x + v_y w_y \).
Substitute the components into the dot product formula: \( 3 \times (-4) + 0 \times 0 \).
Evaluate the expression to check if the dot product equals zero; if it does, the vectors are orthogonal, otherwise they are not.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Dot Product
The dot product of two vectors is a scalar calculated by multiplying corresponding components and summing the results. For vectors v and w, it is v · w = v₁w₁ + v₂w₂ + ... . It measures how much one vector extends in the direction of another.
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05:40
Introduction to Dot Product
Orthogonality of Vectors
Two vectors are orthogonal if their dot product equals zero. This means they are perpendicular to each other in the vector space, indicating no directional overlap.
Recommended video:
03:48Introduction to Vectors
Vector Components and Notation
Vectors are expressed in terms of unit vectors i, j, k representing the x, y, and z axes. Understanding how to interpret and manipulate these components is essential for calculating the dot product and analyzing vector relationships.
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06:01i & j Notation
Related Practice
Textbook Question
In Exercises 27–30, let v = i - 5j and w = -2i + 7j. Find each specified vector or scalar.
||-2v||
Textbook Question
In Exercises 27–30, let v = i - 5j and w = -2i + 7j. Find each specified vector or scalar.
w - v
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Textbook Question
In Exercises 25–30, use Heron's formula to find the area of each triangle. Round to the nearest square unit.
a = 11 yards, b = 9 yards, c = 7 yards
Textbook Question
In Exercises 17–32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively.
a = 7, b = 28, A = 12°
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Textbook Question
In Exercises 17–32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively.
a = 95, c = 125, A = 49°
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