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.

Image courtesy of Cuemath

**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.

Image courtesy of Wolfram Demostration Project

**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:**

**Direction Vectors:**$L5$ has $(1, 0, 1)$, $L6$ has $(0, 1, 1)$.**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.

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.