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Vectors are powerful tools in geometry that help us describe lines efficiently. When we talk about the vector equation of a line, we refer to an expression that uses vectors to define every point on a line. The general form is:

• \( \mathbf{r} = \mathbf{r_0} + t \mathbf{v} \)

Here, \( \mathbf{r} \) represents the position vector of a point on the line, \( \mathbf{r_0} \) is the position vector of a given point through which the line passes, and \( \mathbf{v} \) is the direction vector of the line. The parameter \( t \) can take any real number value, allowing us to trace the entire line as we change \( t \).

In the case of the original exercise, the line passes through point \( P(1, 1, 1) \). Therefore, the vector equation is derived as:

• \( (x, y, z) = (1, 1, 1) + t(1, -1, 1) \)

This equation states that for every value of \( t \), you can find a corresponding point on the line by substituting \( t \) to get the coordinates \( (x, y, z) \).

A direction vector is key to understanding how a line extends through space. It tells us the orientation of the line, enabling it to be identified uniquely. For a line to be defined in three-dimensional space, a direction vector provides a direction from a given point and determines how we "move" along the line.
The direction vector is denoted as \( \mathbf{v} = (a, b, c) \) in vector equations. It defines the slope or direction in which the line stretches infinitely. In our exercise, \( \mathbf{v} = (1, -1, 1) \), showing that

• the line increases by 1 unit in the x-direction,
• decreases by 1 unit in the y-direction, and
• increases by 1 unit in the z-direction per unit of \( t \).

Understanding and correctly interpreting this vector is crucial for plotting or visualizing the line accurately in space.

Coordinate geometry, also known as analytical geometry, provides a link between algebra and geometry through graphs, coordinate points, and equations. This method involves using coordinates to precisely describe geometric figures and their properties.
In three-dimensional coordinate geometry, a point in space is represented as \( (x, y, z) \), and a line can be described using parametric equations derived from a vector equation. For example:

• The parametric equations derived in the exercise are \( x = 1 + t \), \( y = 1 - t \), and \( z = 1 + t \).

These equations allow us to compute the position of points along the line for any value of \( t \), thus providing a dynamic tool to describe lines.
The foundational concept of coordinate geometry assists in solving various geometric problems, analyzing shapes, and understanding spatial relationships efficiently.