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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 25
Chapter 4, Problem 25

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

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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 \( \mathbf{v} - \mathbf{u} \) is performed by subtracting the corresponding components of \( \mathbf{u} \) from \( \mathbf{v} \).
Subtract the \( \mathbf{i} \)-components: \( -3 - 2 = -5 \).
Subtract the \( \mathbf{j} \)-components: \( 7 - (-5) = 7 + 5 = 12 \).
Write the resulting vector as \( \mathbf{v} - \mathbf{u} = -5\mathbf{i} + 12\mathbf{j} \).

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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 v and u, v - u 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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Adding Vectors Geometrically

Unit Vectors i and j

Unit vectors i and j represent the standard basis vectors along the x-axis and y-axis, respectively. They have a magnitude of one and direction along their 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 25–26, let v be the vector from initial point P₁ to terminal point P₂. Write v in terms of i and j.P₁ = (2, -1), P₂ = (5, -3)
Textbook Question

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

v = 2i + 8j, w = 4i - j

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 = 30, b = 40, A = 20°

Textbook Question

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

v = -3j

Textbook Question

In Exercises 25–30, use Heron's formula to find the area of each triangle. Round to the nearest square unit.

a = 4 feet, b = 4 feet, c = 2 feet

Textbook Question

In Exercises 25–29, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [4(cos 50° + i sin 50°)]³