Textbook Question
Find the dot product for each pair of vectors.
4i, 5i - 9j
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8. Vectors
Geometric Vectors
Problem 7.43a
Textbook Question
Given vectors u and v, find: 2u.
u = 〈-1, 2〉, v = 〈3, 0〉
Verified step by step guidance1
Identify the vector u, which is given as \( \langle -1, 2 \rangle \).
Understand that the problem asks for 2u, which means you need to multiply the vector u by the scalar 2.
To multiply a vector by a scalar, multiply each component of the vector by the scalar.
Multiply the first component of u, which is -1, by 2 to get the new first component.
Multiply the second component of u, which is 2, by 2 to get the new second component.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Operations
Vector operations include addition, subtraction, and scalar multiplication. Scalar multiplication involves multiplying a vector by a scalar (a real number), which scales the vector's magnitude without changing its direction. For example, multiplying vector u = 〈-1, 2〉 by 2 results in the vector 〈-2, 4〉.
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Vector Representation
Vectors are represented as ordered pairs or tuples in a coordinate system, indicating their position in space. In this case, u = 〈-1, 2〉 means the vector starts at the origin and points to the coordinates (-1, 2). Understanding this representation is crucial for performing operations like scalar multiplication.
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Geometric Interpretation of Vectors
Vectors can be interpreted geometrically as arrows in a coordinate plane, where the direction and length represent the vector's orientation and magnitude, respectively. When performing operations like scalar multiplication, the resulting vector's length changes, but its direction remains the same if the scalar is positive, which is essential for visualizing the outcome.
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