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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 23
Chapter 4, Problem 23

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

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Identify the given vectors: \( \mathbf{u} = 2\mathbf{i} - 5\mathbf{j} \) and \( \mathbf{v} = -3\mathbf{i} + 7\mathbf{j} \).
Recall that vector subtraction \( \mathbf{u} - \mathbf{v} \) is performed by subtracting the corresponding components of \( \mathbf{v} \) from \( \mathbf{u} \).
Subtract the \( \mathbf{i} \)-components: \( 2 - (-3) = 2 + 3 \).
Subtract the \( \mathbf{j} \)-components: \( -5 - 7 = -5 - 7 \).
Write the resulting vector as \( (2 + 3)\mathbf{i} + (-5 - 7)\mathbf{j} \), which is the vector \( \mathbf{u} - \mathbf{v} \).

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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. Understanding this form allows for straightforward vector operations like addition and subtraction.
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Position Vectors & Component Form

Vector Subtraction

Vector subtraction involves subtracting corresponding components of two vectors. For vectors u and v, u - v is found by subtracting the x-components and y-components separately, resulting in a new vector. This operation is essential for finding the difference or relative position between vectors.
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Unit Vectors i and j

Unit vectors i and j represent the standard basis vectors along the x and y axes, respectively. They have a magnitude of one and direction along their respective axes. Expressing vectors in terms of i and j simplifies calculations and visualization in the plane.
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i & j Notation
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 = 6.1, b = 4, A = 162°
Textbook Question
In Exercises 22–24, sketch each vector as a position vector and find its magnitude.v = -3i - 4j
Textbook Question

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

v = i + j, w = i - j

Textbook Question

In Exercises 9–24, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree.

a = 63, b = 22, c = 50

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 = 10, b = 30, A = 150°

Textbook Question

In Exercises 22–24, sketch each vector as a position vector and find its magnitude.

v = -3j