Coincident: A\u2081B\u2082 \u2212 A\u2082B\u2081 = 0 and C also proportional
det = (A\u2081B\u2082\u2212A\u2082B\u2081) is the cross product of normals, dot = (A\u2081A\u2082+B\u2081B\u2082) is the dot product.
⚠If A\u2081B\u2082\u2212A\u2082B\u2081 \u2260 0, the lines must intersect. If 0 and C is also proportional, the lines are coincident.
Line Relationship Details
Two lines in a plane have only three possible relationships: parallel (including coincident), perpendicular, or intersecting. Coefficient comparison precisely distinguishes these cases.
Parallel
Normal vectors (A,B) are parallel (proportional), slopes are equal. Parallel lines never intersect.
Perpendicular
Normal vectors dot product = 0: A\u2081A\u2082+B\u2081B\u2082=0. Works for all lines including vertical.
Intersecting
When A\u2081B\u2082\u2212A\u2082B\u2081 \u2260 0, the lines are not parallel and must intersect at exactly one point.
Coincident
When all coefficients are proportional, the two lines are identical, a special case of parallel.
Teaching Example: L\u2081: 2x\u2212y\u22123=0, L\u2082: 2x\u2212y\u22125=0. det=2\u00d7(\u22121)\u22122\u00d7(\u22121)=0, so parallel (not coincident). If L\u2082: 4x\u22122y\u22126=0, they would be coincident.
General form: A\u2081B\u2082\u2212A\u2082B\u2081=0 and (A\u2081C\u2082\u2212A\u2082C\u2081)\u00b2+(B\u2081C\u2082\u2212B\u2082C\u2081)\u00b2\u22600 means parallel but not coincident. All coefficients proportional means coincident.
How to determine if two lines are perpendicular?▼
A\u2081A\u2082+B\u2081B\u2082=0 means perpendicular. The dot product of normal vectors is zero. Works regardless of slope existence.
What is the relationship between general form and parallel/perpendicular?▼
Normal vector (A,B), direction vector (\u2212B,A). Parallel: normals proportional. Perpendicular: dot product A\u2081A\u2082+B\u2081B\u2082=0.
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