Textbook Question
Given vectors u and v, find: 2u + 3v.
u = 2i, v = i + j
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8. Vectors
Geometric Vectors
Problem 43c
Textbook Question
Given vectors u and v, find: v - 3u.
u = 〈-1, 2〉, v = 〈3, 0〉
Verified step by step guidance1
Identify the given vectors: \( \mathbf{u} = \langle -1, 2 \rangle \) and \( \mathbf{v} = \langle 3, 0 \rangle \).
Understand that the expression \( \mathbf{v} - 3\mathbf{u} \) means you need to multiply vector \( \mathbf{u} \) by the scalar 3, then subtract the resulting vector from \( \mathbf{v} \).
Calculate the scalar multiplication: multiply each component of \( \mathbf{u} \) by 3, which gives \( 3 \mathbf{u} = \langle 3 \times (-1), 3 \times 2 \rangle = \langle -3, 6 \rangle \).
Perform the vector subtraction by subtracting the corresponding components of \( 3\mathbf{u} \) from \( \mathbf{v} \): \( \mathbf{v} - 3\mathbf{u} = \langle 3 - (-3), 0 - 6 \rangle \).
Simplify the subtraction inside the components to get the resulting vector.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Representation
Vectors are quantities defined by both magnitude and direction, often represented as ordered pairs or tuples in coordinate form, such as u = 〈x, y〉. Understanding how to interpret these components is essential for performing operations like addition, subtraction, and scalar multiplication.
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Scalar Multiplication of Vectors
Scalar multiplication involves multiplying each component of a vector by a real number (scalar). For example, multiplying vector u = 〈x, y〉 by scalar 3 results in 〈3x, 3y〉. This operation changes the vector's magnitude but not its direction unless the scalar is negative.
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Vector Addition and Subtraction
Adding or subtracting vectors is done component-wise: for vectors a = 〈a1, a2〉 and b = 〈b1, b2〉, a ± b = 〈a1 ± b1, a2 ± b2〉. This principle allows combining vectors or finding the difference between them, which is crucial for solving expressions like v - 3u.
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