91Ó°ÊÓ

When dealing with vectors and lines, a direction vector plays a key role. It is essentially a vector that indicates the direction in which a line is pointing. The direction vector is derived from two points on a line. By subtracting the coordinates of the initial point from those of the terminal point, you get the direction vector.
For instance, given two points, \( P = (4, -1, 3) \) and \( Q = (2, 1, 3) \), the direction vector \( \overrightarrow{PQ} \) is calculated as follows:

• Subtract the coordinates of \( P \) from \( Q \): \( (2 - 4, 1 - (-1), 3 - 3) \).
• This results in the vector \( (-2, 2, 0) \).

The vector \( (-2, 2, 0) \) indicates the direction from point \( P \) to point \( Q \). It captures both the change in position and the orientation of the line in space.

In the context of vector equations of a line, the scalar parameter is typically denoted by \( t \). This variable gives you the flexibility to represent all the points on the line by substituting different values.
The vector equation can be expressed as \( \mathbf{r} = \mathbf{a} + t \mathbf{b} \), where:

• \( \mathbf{a} \) is a known point on the line.
• \( \mathbf{b} \) is the direction vector.

By varying \( t \):

• When \( t = 0 \), you find yourself at \( \mathbf{a} \), the starting point of the line.
• As \( t \) changes, different points along the line are obtained, moving in the direction of \( \mathbf{b} \).

This parameter concept is vital because it helps to flexibly describe an endless number of points along the line. It connects the arithmetic of the line’s direction with the graphical intuition of its path.

The vector equation of a line provides a straightforward method to describe a line through two points in space. When you have two points, you inherently establish a line that connects them.
For instance, you are given the points \( P = (4, -1, 3) \) and \( Q = (2, 1, 3) \). To form the vector equation of the line:

• The starting point on the line, \( \mathbf{a} \), can be point \( P \).
• You use the direction vector \( \overrightarrow{PQ} = (-2, 2, 0) \) derived from \( P \) and \( Q \).

You then substitute these into the vector equation template as follows:
\[ \mathbf{r}(t) = (4, -1, 3) + t(-2, 2, 0) \]
This equation concisely represents all points on the line determined by moving from the point \( P \) in the orientation and magnitude specified by the direction vector, through varying \( t \). It's a powerful illustration of connecting algebraic calculations with geometric visualization.