Is it the correct way to prove that if two lines are parallel, then θv + βw ≠ 0

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Is it the correct way to prove that if two lines are parallel, then  θv + βw ≠ 0
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Proving ( \theta v + \beta w \neq 0 ) for Parallel Lines: A Vector-Based Approach

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 approach to proving that if two lines are parallel, then ( \theta v + \beta w \neq 0 ), is conceptually correct in its structure but needs some clarification in the explanation. Here’s a breakdown of your proof with some corrections and improvements for clarity:

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Proof: If Two Lines Are Parallel, Then ( \theta v + \beta w \neq 0 )

Step 1: Representing the Lines

We are given two parallel lines:
1. ( x + y = 2 ), or equivalently ( x + y – 2 = 0 ). Label this as ( v ).
2. ( 2x + 2y = 4 ), or equivalently ( 2x + 2y – 4 = 0 ). Label this as ( w ).

Step 2: Reducing to Direction Vectors

For parallel lines, the direction (or vector component) is the same, and constant terms do not affect the vector direction. Removing constants gives us:
1. ( v = x + y = 0 ).
2. ( w = 2(x + y) = 0 ), which is proportional to ( v ). Specifically, ( w = 2v ).

Step 3: Expressing ( \theta v + \beta w )

Using the given setup, let:
[
\theta v + \beta w = \theta(x + y) + \beta[2(x + y)].
]
Simplify:
[
\theta v + \beta w = (\theta + 2\beta)(x + y).
]

Step 4: Parallel Lines Imply Non-Zero Coefficients

For ( \theta v + \beta w = 0 ), the scalar ( (\theta + 2\beta) ) must equal zero. That implies:
[
\theta + 2\beta = 0 \quad \text{(i.e., ( \theta ) and ( \beta ) must cancel each other)}.
]

However, since ( v ) and ( w ) are proportional (( w = 2v )), the only way this can happen is if both ( \theta ) and ( \beta ) are zero. But if ( \theta = 0 ) and ( \beta = 0 ), the expression ( \theta v + \beta w = 0 ) holds trivially, which contradicts the assumption that ( \theta ) and ( \beta ) are non-zero scalars.

Thus, for parallel lines with non-zero scalars ( \theta ) and ( \beta ), ( \theta v + \beta w \neq 0 ).

Step 5: Conclusion

This demonstrates that for two parallel lines, ( \theta v + \beta w \neq 0 ), as the proportional relationship between ( v ) and ( w ) ensures that their scalar combination does not sum to zero unless both coefficients are zero.

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Key Insights

• Removing constants simplifies the problem to the direction vectors, which are unaffected by shifts in the constant term.
• The proportionality between ( v ) and ( w ) ensures that the scalar combination ( \theta v + \beta w \neq 0 ) unless both coefficients are zero.
• Parallelism is preserved in the vector form since the directions ( v ) and ( w ) remain aligned.

Suggestions for Improvement

• Clearly distinguish between direction vectors and lines to avoid confusion.
• Explicitly state the significance of proportionality between ( v ) and ( w ).

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