Textbook Question
Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary.
〈-4, 4√3〉
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8. Vectors
Geometric Vectors
Problem 34
Textbook Question
Given u = 〈-2, 5〉 and v = 〈4, 3〉, find each of the following.
v - u
Verified step by step guidance1
Identify the given vectors: \( \mathbf{u} = \langle -2, 5 \rangle \) and \( \mathbf{v} = \langle 4, 3 \rangle \).
Recall that vector subtraction \( \mathbf{v} - \mathbf{u} \) is performed component-wise: subtract the corresponding components of \( \mathbf{u} \) from \( \mathbf{v} \).
Set up the subtraction for each component: \( (v_x - u_x, v_y - u_y) \), where \( v_x = 4, u_x = -2, v_y = 3, u_y = 5 \).
Calculate each component difference separately: \( 4 - (-2) \) for the x-component and \( 3 - 5 \) for the y-component.
Combine the results to write the vector \( \mathbf{v} - \mathbf{u} \) as \( \langle 4 - (-2), 3 - 5 \rangle \).
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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 two dimensions. For example, u = 〈-2, 5〉 indicates a vector with components -2 along the x-axis and 5 along the y-axis.
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Vector Subtraction
Vector subtraction involves subtracting corresponding components of two vectors. For vectors v = 〈v₁, v₂〉 and u = 〈u₁, u₂〉, the difference v - u is 〈v₁ - u₁, v₂ - u₂〉, resulting in a new vector.
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Component-wise Operations
Operations on vectors such as addition and subtraction are performed component-wise, meaning each component is handled independently. This simplifies calculations and helps visualize vector operations geometrically.
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