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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 33
Chapter 4, Problem 33

In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.
3v - 4w

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Identify the given vectors: \( \mathbf{v} = -3\mathbf{i} + 7\mathbf{j} \) and \( \mathbf{w} = -\mathbf{i} - 6\mathbf{j} \).
Multiply vector \( \mathbf{v} \) by the scalar 3: calculate \( 3\mathbf{v} = 3(-3\mathbf{i} + 7\mathbf{j}) \).
Multiply vector \( \mathbf{w} \) by the scalar 4: calculate \( 4\mathbf{w} = 4(-\mathbf{i} - 6\mathbf{j}) \).
Subtract the vector \( 4\mathbf{w} \) from \( 3\mathbf{v} \): compute \( 3\mathbf{v} - 4\mathbf{w} \) by subtracting corresponding components.
Write the resulting vector in component form \( a\mathbf{i} + b\mathbf{j} \) by combining the \( \mathbf{i} \) and \( \mathbf{j} \) components from the subtraction.

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Key Concepts

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

Vector Representation in Component Form

Vectors in two dimensions can be expressed as components along the i (x-axis) and j (y-axis) unit vectors. For example, u = 2i - 5j means the vector has an x-component of 2 and a y-component of -5. This form allows for straightforward addition, subtraction, and scalar multiplication of vectors.
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Scalar Multiplication of Vectors

Scalar multiplication involves multiplying each component of a vector by a scalar (a real number). For instance, multiplying vector v by 3 means multiplying both its i and j components by 3, resulting in a new vector scaled in magnitude but with the same direction if the scalar is positive.
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Vector Addition and Subtraction

Adding or subtracting vectors is done component-wise by adding or subtracting their corresponding i and j components. For example, to find 3v - 4w, first multiply each vector by its scalar, then subtract the resulting components to get the final vector.
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Related Practice
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 = 1.4, b = 2.9, A = 142°

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Textbook Question

In Exercises 33–38, find the area of the triangle having the given measurements. Round to the nearest square unit.

A = 22°, b = 20 feet, c = 50 feet

Textbook Question

In Exercises 33–38, find projᵥᵥ v. Then decompose v into two vectors, v₁ and v₂, where v₁ is parallel to w and v₂ is orthogonal to w. v = 3i - 2j, w = i - j

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Textbook Question

In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.

||2u||

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Textbook Question

In Exercises 31–32, find the unit vector that has the same direction as the vector v.

v = -i + 2j

Textbook Question

In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.

3w + 2v

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