In this comprehensive exploration of the equation of a line in vector terms, we delve into the intricacies of vector algebra and its application in defining lines in both two and three-dimensional spaces. This knowledge is fundamental for students in understanding complex geometrical relationships and solving advanced mathematical problems.

Vector Equation of a Line

The vector equation of a line in space is expressed as $\mathbf{r} = \mathbf{a} + t\mathbf{b}$. This equation is pivotal in vector algebra, representing a line in terms of vectors. Here:

Image courtesy of Cuemath

Finding a Line's Equation:

1. Position Vectors: Start with position vectors of two points, $\mathbf{a}$ and $\mathbf{b}$.

2. Direction Vector: Use the difference $\mathbf{b} - \mathbf{a}$ as the direction vector.

3. Line Equation: Write the equation as $\mathbf{r} = \mathbf{a} + t(\mathbf{b} - \mathbf{a})$, with $t$ varying to trace the line.

Examples

Example 1: Finding a Point on a Line

Given the line $\mathbf{r} = 3\mathbf{i} + 4\mathbf{j} + t(2\mathbf{i} - \mathbf{j} + 3\mathbf{k})$, find the position vector of a point on the line when $t = 2$.

Solution:

1. Substitute $t = 2$ into the equation.

2. Calculate the position vector: $\mathbf{r} = 3\mathbf{i} + 4\mathbf{j} + 2(2\mathbf{i} - \mathbf{j} + 3\mathbf{k}) = 7\mathbf{i} + 2\mathbf{j} + 6\mathbf{k}$.

3. The position vector of the point is $7\mathbf{i} + 2\mathbf{j} + 6\mathbf{k}$.

Example 2: Equation of a Line Through Two Points

Find the equation of the line passing through the points with position vectors $2\mathbf{i} + 3\mathbf{j}$ and $-\mathbf{i} + 4\mathbf{j} + 5\mathbf{k}.$

Solution:

1. Let $\mathbf{a} = 2\mathbf{i} + 3\mathbf{j}$ and $\mathbf{b} = -\mathbf{i} + 4\mathbf{j} + 5\mathbf{k}$.

2. The equation of the line is $\mathbf{r} = \mathbf{a} + t(\mathbf{b} - \mathbf{a}) = 2\mathbf{i} + 3\mathbf{j} + t(-3\mathbf{i} + \mathbf{j} + 5\mathbf{k})$.

3. This represents all points on the line as a linear combination of $\mathbf{a}$ and $\mathbf{b}$.