Textbook Question
Write each vector in the form a i + b j.
〈2, 0〉
8. Vectors
Geometric Vectors
Problem 41b
Textbook Question
Given vectors u and v, find: 2u + 3v.
u = 2i, v = i + j
Verified step by step guidance1
Identify the given vectors: \( \mathbf{u} = 2\mathbf{i} \) and \( \mathbf{v} = \mathbf{i} + \mathbf{j} \).
Multiply vector \( \mathbf{u} \) by 2: calculate \( 2\mathbf{u} = 2 \times 2\mathbf{i} \).
Multiply vector \( \mathbf{v} \) by 3: calculate \( 3\mathbf{v} = 3 \times (\mathbf{i} + \mathbf{j}) \).
Add the resulting vectors from the previous two steps: \( 2\mathbf{u} + 3\mathbf{v} \).
Combine like terms (i.e., the \( \mathbf{i} \) components together and the \( \mathbf{j} \) components together) to express the final vector in terms of \( \mathbf{i} \) and \( \mathbf{j} \).
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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 can be expressed as sums of their components along standard unit vectors, typically i and j in two dimensions. For example, u = 2i means the vector has a component 2 along the x-axis and 0 along the y-axis. 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 u by 2 scales its magnitude by 2 without changing its direction. This operation is essential for combining vectors with different weights, as in 2u + 3v.
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Vector Addition
Vector addition is performed by adding corresponding components of the vectors. For example, adding vectors a = a₁i + a₂j and b = b₁i + b₂j results in (a₁ + b₁)i + (a₂ + b₂)j. This principle is used to find the resultant vector 2u + 3v by first scaling and then adding the vectors.
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