What is vector equation of the line

Answer

To find the vector equation of a line in three-dimensional space, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Points**:
- Let \( \mathbf{a} \) be the position vector of a point on the line. This point can be represented as \( (x_0, y_0, z_0) \) in Cartesian coordinates. Thus, \( \mathbf{a} = x_0 \mathbf{i} + y_0 \mathbf{j} + z_0 \mathbf{k} \).

2. **Determine the Direction Vector**:
- Let \( \mathbf{b} \) be the direction vector of the line. This vector indicates the direction in which the line extends. If the direction vector is given by \( (a, b, c) \), then \( \mathbf{b} = a \mathbf{i} + b \mathbf{j} + c \mathbf{k} \).

3. **Construct the Vector Equation**:
- The vector equation of the line can be expressed as:
\[ \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \]
- Here, \( \mathbf{r} \) is the position vector of any point on the line, \( \lambda \) is a scalar parameter that varies along the line, \( \mathbf{a} \) is the position vector of a specific point on the line, and \( \mathbf{b} \) is the direction vector.

4. **Interpret the Equation**:
- As \( \lambda \) takes on different values, \( \mathbf{r} \) traces out all points on the line. When \( \lambda = 0 \), \( \mathbf{r} \) equals \( \mathbf{a} \), which is a point on the line. As \( \lambda \) increases or decreases, \( \mathbf{r} \) moves along the line in the direction of \( \mathbf{b} \).

### Final Answer:
The vector equation of the line is given by:
\[ \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \]