Vector analysis is essential for determining the spatial relationships between lines in both two and three-dimensional geometries. This section examines the conditions and methods for identifying parallel, intersecting, and skew lines, providing a step-by-step approach to understanding their interactions.

Parallel Lines

Parallel lines are two or more lines in a plane that never intersect. They have the same slope but different y-intercepts in 2D geometry.

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Example:

Consider two lines on a plane, $L1: y = 2x + 3$ and $L2: y = 2x - 4$.

Solution:

1. Slopes: From $y = mx + b$, slope $m = 2$ for both $L1$ and $L2$.

2. Comparison: Equal slopes mean $L1$ and $L2$ are parallel.

Conclusion: $L1$and $L2$ are parallel due to identical slopes.

Intersecting Lines

Intersecting lines cross at a single point. In 2D geometry, this occurs when two lines have different slopes.

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Example:

Two lines, $L3: y = x + 1$ and $L4: y = -2x + 3$.

Solution:

1. Equation: Set $x + 1 = -2x + 3$ to find the intersection.

2. Solve for $x$: Rearrange to $3x = 2$, so $x = \frac{2}{3}.$

3. Solve for $y$: Substitute $x$ into $L3$ to get $y = \frac{5}{3}$.

Conclusion: $L3$ and $L4$ intersect at $\left(\frac{2}{3}, \frac{5}{3}\right)$.

Skew Lines

Skew lines are lines that do not intersect and are not parallel, usually found in 3D geometry.

Image courtesy of Cuemath

Example:

Consider two lines in 3D space, $L5: (x, y, z) = (1, 2, 3) + t(1, 0, 1)$ and $L6: (x, y, z) = (2, 4, 5) + s(0, 1, 1)$, where $t$ and $s$ are parameters.

Solution:

1. Direction Vectors: $L5$ has $(1, 0, 1)$, $L6$ has $(0, 1, 1)$.
2. Not Parallel: Direction vectors aren't scalar multiples.

3. No Intersection: No $t$ and $s$ satisfy both equations simultaneously.

Conclusion: $L5$ and $L6$ are skew, meaning they do not intersect or run parallel.