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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 8
Chapter 4, Problem 8

In Exercises 5–8, let v = -5i + 2j and w = 2i - 4j Find the specified vector, scalar, or angle. projᵥᵥv

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Identify the vectors involved: here, both vectors are \( \mathbf{v} = -5\mathbf{i} + 2\mathbf{j} \) and \( \mathbf{w} = 2\mathbf{i} - 4\mathbf{j} \), but the problem asks for \( \text{proj}_{\mathbf{v}} \mathbf{v} \), which means the projection of \( \mathbf{v} \) onto itself.
Recall the formula for the projection of a vector \( \mathbf{a} \) onto a vector \( \mathbf{b} \): \[ \text{proj}_{\mathbf{b}} \mathbf{a} = \left( \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \right) \mathbf{b} \] where \( \mathbf{a} \cdot \mathbf{b} \) is the dot product of \( \mathbf{a} \) and \( \mathbf{b} \).
Since we are projecting \( \mathbf{v} \) onto itself, substitute \( \mathbf{a} = \mathbf{v} \) and \( \mathbf{b} = \mathbf{v} \) into the formula: \[ \text{proj}_{\mathbf{v}} \mathbf{v} = \left( \frac{\mathbf{v} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \right) \mathbf{v} \]
Calculate the dot product \( \mathbf{v} \cdot \mathbf{v} \) by multiplying corresponding components and summing: \[ \mathbf{v} \cdot \mathbf{v} = (-5)(-5) + (2)(2) \]
Simplify the fraction \( \frac{\mathbf{v} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \) which will equal 1, so the projection \( \text{proj}_{\mathbf{v}} \mathbf{v} \) is simply \( \mathbf{v} \) itself.

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Key Concepts

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

Vector Projection

Vector projection of one vector onto another is the component of the first vector in the direction of the second. It is found by scaling the second vector by the scalar projection, which involves the dot product divided by the magnitude squared of the second vector.
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Dot Product of Vectors

The dot product is an algebraic operation that takes two equal-length sequences of numbers (vectors) and returns a single number. It is calculated as the sum of the products of corresponding components and is used to find angles between vectors and projections.
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Vector Magnitude

The magnitude (or length) of a vector is the distance from the origin to the point represented by the vector in space. It is calculated using the Pythagorean theorem as the square root of the sum of the squares of its components.
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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 9–16, letu = 2i - j, v = 3i + j, and w = i + 4j.Find each specified scalar.u ⋅ (v + w)
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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 1–8, use the given vectors to find v⋅w and v⋅v. v = 5i, w = j

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Textbook Question

In Exercises 5–12, sketch each vector as a position vector and find its magnitude. v = i - j

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.A = 162°, b = 11.2, c = 48.2
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