Textbook Question
Find the dot product for each pair of vectors.
4i, 5i - 9j
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8. Vectors
Geometric Vectors
Problem 43a
Textbook Question
Given vectors u and v, find: 2u.
u = 〈-1, 2〉, v = 〈3, 0〉
Verified step by step guidance1
Identify the vector \( \mathbf{u} = \langle -1, 2 \rangle \) and the scalar multiplier, which is 2 in this case.
Recall that multiplying a vector by a scalar means multiplying each component of the vector by that scalar.
Apply the scalar multiplication to vector \( \mathbf{u} \): multiply the first component by 2 and the second component by 2.
Write the resulting vector as \( 2\mathbf{u} = \langle 2 \times (-1), 2 \times 2 \rangle \).
Simplify the components to express the final vector in the form \( \langle a, b \rangle \), where \( a \) and \( b \) are the products from the previous step.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Scalar Multiplication
Scalar multiplication involves multiplying each component of a vector by a scalar (a real number). For example, multiplying vector u = 〈x, y〉 by 2 results in 2u = 〈2x, 2y〉. This operation scales the vector's magnitude without changing its direction.
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Vector Notation and Components
Vectors in two dimensions are represented as ordered pairs 〈x, y〉, where x and y are components along the horizontal and vertical axes. Understanding this notation is essential for performing operations like addition, subtraction, and scalar multiplication.
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Basic Vector Operations
Basic vector operations include addition, subtraction, and scalar multiplication. These operations follow component-wise rules, allowing manipulation of vectors algebraically. Mastery of these operations is fundamental for solving vector-related problems.
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