Why lines are set to pass through the origin: Understanding ax+by=0 and vector preservation

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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.

Yes, your understanding is correct! Here’s a detailed explanation:

Why Equations Are Set to Zero:

When a line equation is written in the form ( ax + by = 0 ), it is intentionally crafted to ensure that the line passes through the origin ((0, 0)).

This form highlights:
– The vector direction ((a, b)), which represents the slope or orientation of the line.
– Setting the equation to zero removes any constant term (e.g., ( c ) in ( ax + by + c = 0 )) that might shift the line away from the origin.

Any Line Can Pass Through the Origin:

Yes, any straight line in two dimensions can be made to pass through the origin by setting ( c = 0 ) in its general equation ( ax + by + c = 0 ). For example:

• The line ( 9x + 10y = c ) passes through the origin when ( c = 0 ), resulting in ( 9x + 10y = 0 ).
• The vector ( (9, 10) ), which is perpendicular to the line, remains the same because only the position of the line changes, not its orientation.

Preserving the Line’s Vector:

The “vector” here refers to the direction vector of the line (proportional to ((-b, a)) for ( ax + by = 0)). When you set ( c = 0 ):
1. The orientation of the line does not change because the coefficients ( a ) and ( b ) remain unchanged.
2. Only the intercept changes, aligning the line to pass through the origin.

Practical Implication:

This approach is often used in vector algebra and linear transformations because it simplifies calculations:
– A line passing through the origin directly corresponds to a subspace in vector space.
– Equations like ( ax + by = 0 ) are useful in geometric interpretations and linear mappings.

So yes, rewriting any line equation to pass through the origin, such as converting ( 9x + 10y = c ) into ( 9x + 10y = 0 ), maintains its directional properties (its vector) while changing its position.

 

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