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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 31
Chapter 4, Problem 31

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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Identify the given vectors: \( \mathbf{u} = 2\mathbf{i} - 5\mathbf{j} \), \( \mathbf{v} = -3\mathbf{i} + 7\mathbf{j} \), and \( \mathbf{w} = -\mathbf{i} - 6\mathbf{j} \).
Calculate the scalar multiplication of vector \( \mathbf{w} \) by 3: multiply each component of \( \mathbf{w} \) by 3, resulting in \( 3\mathbf{w} = 3(-\mathbf{i}) + 3(-6\mathbf{j}) \).
Calculate the scalar multiplication of vector \( \mathbf{v} \) by 2: multiply each component of \( \mathbf{v} \) by 2, resulting in \( 2\mathbf{v} = 2(-3\mathbf{i}) + 2(7\mathbf{j}) \).
Add the resulting vectors from the previous two steps component-wise: add the \( \mathbf{i} \) components together and the \( \mathbf{j} \) components together to find \( 3\mathbf{w} + 2\mathbf{v} \).
Write the final vector in the form \( a\mathbf{i} + b\mathbf{j} \), where \( a \) and \( b \) are the sums of the respective components.

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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 a combination of unit vectors i and j, representing the x and y components respectively. For example, u = 2i - 5j means the vector has an x-component of 2 and a y-component of -5. Understanding this form allows for straightforward vector addition and scalar multiplication.
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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 w 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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Multiplying Vectors By Scalars

Vector Addition

Vector addition is performed by adding corresponding components of two vectors. For example, adding 2v and 3w requires adding the x-components of 2v and 3w together, and similarly for the y-components, resulting in a new vector that combines both directions and magnitudes.
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Adding Vectors Geometrically
Related Practice
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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Textbook Question

In Exercises 23–32, use the dot product to determine whether v and w are orthogonal.

v = 3i, w = -4j

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

v = -i + 2j

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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