Textbook Question
Two forces act on a point in the plane. The angle between the two forces is given. Find the magnitude of the resultant force.
forces of 116 and 139 lb, forming an angle of 140° 50′
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8. Vectors
Geometric Vectors
Problem 7.34
Textbook Question
Given u = 〈-2, 5〉 and v = 〈4, 3〉, find each of the following.
v - u
Verified step by step guidance1
Identify the components of vector \( u \) as \( u = \langle -2, 5 \rangle \) and vector \( v \) as \( v = \langle 4, 3 \rangle \).
To find \( v - u \), subtract the corresponding components of vector \( u \) from vector \( v \).
Subtract the first component of \( u \) from the first component of \( v \): \( 4 - (-2) \).
Subtract the second component of \( u \) from the second component of \( v \): \( 3 - 5 \).
Combine the results from the previous steps to form the resulting vector: \( \langle 4 - (-2), 3 - 5 \rangle \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Subtraction
Vector subtraction involves finding the difference between two vectors by subtracting their corresponding components. For vectors u = 〈u1, u2〉 and v = 〈v1, v2〉, the result of v - u is given by 〈v1 - u1, v2 - u2〉. This operation is fundamental in vector analysis and is used to determine the relative position of one vector to another.
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Component Form of Vectors
Vectors can be expressed in component form, which represents them as ordered pairs or triples. In this case, u = 〈-2, 5〉 and v = 〈4, 3〉 are two-dimensional vectors where the first component indicates the horizontal direction and the second component indicates the vertical direction. Understanding component form is essential for performing operations like addition and subtraction.
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Coordinate System
A coordinate system provides a framework for locating points in space using ordered pairs or triples. In a two-dimensional Cartesian coordinate system, the x-axis represents the horizontal direction and the y-axis represents the vertical direction. This system is crucial for visualizing vectors and performing operations such as vector subtraction, as it allows for a clear understanding of the vectors' positions and directions.
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