Textbook Question
Given vectors u and v, find: 2u + 3v.
u = 〈-1, 2〉, v = 〈3, 0〉
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8. Vectors
Geometric Vectors
Problem 41a
Textbook Question
Given vectors u and v, find: 2u.
u = 2i, v = i + j
Verified step by step guidance1
Identify the given vector \( \mathbf{u} \). Here, \( \mathbf{u} = 2\mathbf{i} \), which means the vector has a component of 2 in the \( \mathbf{i} \) (x) direction and 0 in the \( \mathbf{j} \) (y) direction.
Understand that multiplying a vector by a scalar (in this case, 2) means multiplying each component of the vector by that scalar.
Write the scalar multiplication operation: \( 2\mathbf{u} = 2 \times (2\mathbf{i}) \).
Multiply the scalar 2 by each component of \( \mathbf{u} \). Since \( \mathbf{u} = 2\mathbf{i} + 0\mathbf{j} \), this becomes \( 2 \times 2\mathbf{i} + 2 \times 0\mathbf{j} \).
Simplify the expression to get the resulting vector \( 2\mathbf{u} \).
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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 a combination of unit vectors along coordinate axes, such as i and j in two dimensions. For example, u = 2i means the vector has a magnitude of 2 units along the x-axis and zero along the y-axis.
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Scalar Multiplication of Vectors
Multiplying a vector by a scalar involves multiplying each component of the vector by that scalar. This operation changes the vector's magnitude but not its direction unless the scalar is negative, which reverses the direction.
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Vector Addition and Subtraction
Vectors can be added or subtracted by combining their corresponding components. Although not directly required here, understanding vector addition helps in comprehending vector operations and their geometric interpretations.
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