Textbook Question
Given vectors u and v, find: v - 3u.
u = 〈-1, 2〉, v = 〈3, 0〉
Ch. 7 - Applications of Trigonometry and Vectors
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 8, Problem 43b
Given vectors u and v, find: 2u + 3v.
u = 〈-1, 2〉, v = 〈3, 0〉
Verified step by step guidance1
Identify the given vectors: \( \mathbf{u} = \langle -1, 2 \rangle \) and \( \mathbf{v} = \langle 3, 0 \rangle \).
Multiply vector \( \mathbf{u} \) by the scalar 2: calculate \( 2\mathbf{u} = 2 \times \langle -1, 2 \rangle \).
Multiply vector \( \mathbf{v} \) by the scalar 3: calculate \( 3\mathbf{v} = 3 \times \langle 3, 0 \rangle \).
Add the resulting vectors from steps 2 and 3 component-wise: \( 2\mathbf{u} + 3\mathbf{v} = \langle 2u_1 + 3v_1, 2u_2 + 3v_2 \rangle \), where \( u_1, u_2 \) and \( v_1, v_2 \) are the components of \( \mathbf{u} \) and \( \mathbf{v} \) respectively.
Write the final vector as the sum of the components found in step 4, which represents \( 2\mathbf{u} + 3\mathbf{v} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Addition
Vector addition involves combining two vectors by adding their corresponding components. For vectors u = 〈u₁, u₂〉 and v = 〈v₁, v₂〉, the sum u + v is 〈u₁ + v₁, u₂ + v₂〉. This operation results in a new vector representing the combined effect of both.
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Scalar Multiplication of Vectors
Scalar multiplication scales a vector by multiplying each of its components by a real number (scalar). For a scalar k and vector u = 〈u₁, u₂〉, the product k*u is 〈k*u₁, k*u₂〉. This changes the vector's magnitude without altering its direction unless the scalar is negative.
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Component-wise Operations in 2D Vectors
Operations on 2D vectors are performed component-wise, meaning each x and y component is handled separately. This approach simplifies calculations like 2u + 3v by first scaling each vector and then adding corresponding components to find the resultant vector.
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Related Practice
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Textbook Question
Two people are carrying a box. One person exerts a force of 150 lb at an angle of 62.4° with the horizontal. The other person exerts a force of 114 lb at an angle of 54.9°. Find the weight of the box.
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Textbook Question
Given vectors u and v, find: 2u.
u = 〈-1, 2〉, v = 〈3, 0〉
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