Ch. 4 - Laws of Sines and Cosines; Vectors

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 4 - Laws of Sines and Cosines; Vectors

Problem 3c

Chapter 4, Problem 3c

In Exercises 1–4, u and v have the same direction. In each exercise: Is u = v? Explain.

Verified step by step guidance

Understand that two vectors \( \mathbf{u} \) and \( \mathbf{v} \) having the same direction means they are scalar multiples of each other. This implies \( \mathbf{u} = k \mathbf{v} \) for some scalar \( k > 0 \).

Recall that for \( \mathbf{u} = \mathbf{v} \) to be true, both the magnitude (length) and direction of \( \mathbf{u} \) and \( \mathbf{v} \) must be exactly the same.

Since \( \mathbf{u} \) and \( \mathbf{v} \) have the same direction, check if their magnitudes are equal by comparing \( |\mathbf{u}| \) and \( |\mathbf{v}| \).

If \( |\mathbf{u}| = |\mathbf{v}| \), then \( k = 1 \) and \( \mathbf{u} = \mathbf{v} \). Otherwise, if \( |\mathbf{u}| \neq |\mathbf{v}| \), then \( \mathbf{u} \neq \mathbf{v} \) even though they point in the same direction.

Summarize your conclusion by stating that having the same direction does not guarantee \( \mathbf{u} = \mathbf{v} \); equality requires both direction and magnitude to be identical.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Vector Direction

Vector direction refers to the orientation of a vector in space, independent of its magnitude. Two vectors have the same direction if they lie along the same line or parallel lines, pointing either the same way or exactly opposite. Understanding direction is crucial to compare vectors beyond just their lengths.

Vector Equality

Two vectors are equal if and only if they have the same magnitude and the same direction. Even if vectors share the same direction, they are not equal unless their lengths are identical. This concept helps determine when vectors represent the same quantity.

Scalar Multiplication of Vectors

Scalar multiplication changes a vector's magnitude without altering its direction, unless the scalar is negative, which reverses the direction. Recognizing how scaling affects vectors is essential to understand when vectors with the same direction differ in magnitude and thus are not equal.

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Related Practice

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 1–4, u and v have the same direction. In each exercise: Find ||u||.

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Textbook Question

Let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and j.

P₁ = (2, -5), P₂ = (-6, 6)

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Textbook Question

Let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and j.

P₁ = (-1, 6), P₂ = (7, -5)

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 1–8, solve each triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree.

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