Understanding scalar combinations of lines and their relation to the origin

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Understanding Scalar Combinations of Lines and Their Relation to the Origin

Disclaimer: This article was created with the assistance of an AI language model and is intended for informational purposes only. Please verify any technical details before implementation.

Your inquiry touches on important geometric and algebraic aspects of vector and line operations. Let’s address both parts of your question:

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1. Is ( \theta(2x + 5y – 15) + \beta(3x + 9y – 44) = 0 ) passing through the origin? Can values of ( \theta ) and ( \beta ) make it pass through the origin?

• For the expression ( \theta(2x + 5y – 15) + \beta(3x + 9y – 44) = 0 ), the “line passing through the origin” depends on the condition that the resulting equation after combining the terms has no constant term.
• That is, the constant terms ( -15 ) (from ( v )) and ( -44 ) (from ( w )) must cancel out in the linear combination.

• Simplifying:
  [
  \theta(2x + 5y – 15) + \beta(3x + 9y – 44) = 0
  ]
  Expands to:
  [
  (2\theta + 3\beta)x + (5\theta + 9\beta)y + (-15\theta – 44\beta) = 0
  ]

• To ensure the equation passes through the origin, the constant term ( -15\theta – 44\beta = 0 ).

• Solving ( -15\theta – 44\beta = 0 ), you can find specific values of ( \theta ) and ( \beta ) that eliminate the constant term, making the resulting equation represent a line passing through the origin.

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2. Why is the right-hand side set to zero?

• Setting the equation to zero (( ax + by + c = 0 )) has practical and mathematical advantages:

1. Origin-Centric Operations: When ( c = 0 ), the line passes through the origin, simplifying vector interpretations and calculations.
2. Ease of Linear Combinations: Zero on the right-hand side allows you to combine lines (or vectors) and apply scalar coefficients (( \theta, \beta )) without introducing additional complexities from constants.
3. Vector Representation: Lines expressed in the form ( ax + by + c = 0 ) can be interpreted as vectors (( v = (a, b) )), which makes geometric operations like dot products and projections easier.

The goal of emphasizing zero is often to standardize the equations for analytical consistency and geometric clarity. In higher-dimensional contexts, it also facilitates solving systems of equations.

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If you aim to ensure that the equation ( \theta v + \beta w = 0 ) represents a line through the origin while keeping ( v ) and ( w ) intact as vectors, the constant terms must cancel appropriately.

 

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