Parallel Lines

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Two lines with direction ratios (a₁, b₁, c₁) and (a₂, b₂, c₂) are parallel if

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1. **Theory**: Two lines are parallel if their direction vectors are in the same or opposite directions.
2. **Proportionality**: If vectors $\vec{v_1} = (a_1, b_1, c_1)$ and $\vec{v_2} = (a_2, b_2, c_2)$ are parallel, then $\vec{v_1} = k\vec{v_2}$ for some scalar $k$.
3. **Expansion**: $a_1 = ka_2, b_1 = kb_2, c_1 = kc_2$.
4. **Condition**: Therefore, $a_1/a_2 = b_1/b_2 = c_1/c_2 = k$. Values must be proportional.

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Parallel Lines

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