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Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 4 - Laws of Sines and Cosines; VectorsProblem 40
Chapter 4, Problem 40

In Exercises 40–41, use the dot product to determine whether v and w are orthogonal.
v = 12i - 8j, w = 2i + 3j

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Recall that two vectors \( \mathbf{v} \) and \( \mathbf{w} \) are orthogonal if and only if their dot product is zero, i.e., \( \mathbf{v} \cdot \mathbf{w} = 0 \).
Write down the components of the vectors: \( \mathbf{v} = 12\mathbf{i} - 8\mathbf{j} \) means \( \mathbf{v} = (12, -8) \), and \( \mathbf{w} = 2\mathbf{i} + 3\mathbf{j} \) means \( \mathbf{w} = (2, 3) \).
Calculate the dot product using the formula \( \mathbf{v} \cdot \mathbf{w} = v_1 w_1 + v_2 w_2 \), where \( v_1, v_2 \) are components of \( \mathbf{v} \) and \( w_1, w_2 \) are components of \( \mathbf{w} \).
Substitute the values: \( \mathbf{v} \cdot \mathbf{w} = (12)(2) + (-8)(3) \).
Evaluate the expression to check if the dot product equals zero. If it does, the vectors are orthogonal; if not, they are not orthogonal.

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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. For vectors v = (v1, v2) and w = (w1, w2), the dot product is v1*w1 + v2*w2.
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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. Checking orthogonality involves computing the dot product and verifying if the result is zero.
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Introduction to Vectors

Vector Components and Notation

Vectors can be expressed in terms of unit vectors i and j representing the x and y directions, respectively. Understanding how to interpret and manipulate these components is essential for performing operations like the dot product.
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i & j Notation
Related Practice
Textbook Question

In Exercises 39–40, find h to the nearest tenth.

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Textbook Question

In Exercises 39–46, find the unit vector that has the same direction as the vector v.

v = 3i - 4j

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 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

Textbook Question

In Exercises 39–46, find the unit vector that has the same direction as the vector v.

v = 6i

Textbook Question

In Exercises 39–42, let u = -i + j, v = 3i - 2j, and w = -5j. Find each specified scalar or vector.

projᵤ (v + w)