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 = (-3, 0), P₂ = (-2, -2)
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Ch. 4 - Laws of Sines and Cosines; Vectors
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
Chapter 4, Problem 27
In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.
5v
Verified step by step guidance1
Identify the given vector \( \mathbf{v} = -3\mathbf{i} + 7\mathbf{j} \).
Understand that multiplying a vector by a scalar means multiplying each component of the vector by that scalar.
Multiply the scalar 5 by the \( \mathbf{i} \)-component of \( \mathbf{v} \): \( 5 \times (-3) = -15 \).
Multiply the scalar 5 by the \( \mathbf{j} \)-component of \( \mathbf{v} \): \( 5 \times 7 = 35 \).
Write the resulting vector as \( 5\mathbf{v} = -15\mathbf{i} + 35\mathbf{j} \).
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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 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 operations like addition, subtraction, and scalar multiplication.
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Position Vectors & Component Form
Scalar Multiplication of Vectors
Scalar multiplication involves multiplying each component of a vector by a scalar (a real number). This operation changes the magnitude of the vector without altering its direction if the scalar is positive, or reverses the direction if the scalar is negative. For instance, 5v means multiplying both components of vector v by 5.
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05:05Multiplying Vectors By Scalars
Vector Notation and Operations
Vectors are often denoted using unit vectors i and j to simplify calculations. Operations like addition, subtraction, and scalar multiplication are performed component-wise. Recognizing how to manipulate vectors in this notation is essential for solving problems involving vector quantities in physics and mathematics.
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06:01i & j Notation
Related Practice
Textbook Question
In Exercises 25–30, use Heron's formula to find the area of each triangle. Round to the nearest square unit.
a = 14 meters, b = 12 meters, c = 4 meters
Textbook Question
In Exercises 27–30, let v = i - 5j and w = -2i + 7j. Find each specified vector or scalar.
w - v
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Textbook Question
In Exercises 23–32, use the dot product to determine whether v and w are orthogonal. v = 2i - 2j, w = -i + 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 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