91Ó°ÊÓ

In the world of vector equations, the direction vector is like the compass that guides us. To better understand the concept, consider two points: point \( P(7, -2, 6) \) and point \( Q(-3, 0, 6) \), which are on a line. The direction vector essentially describes how you travel from one point to the other in three-dimensional space.

To find the direction vector \( \vec{PQ} \), subtract the coordinates of point \( P \) from those of point \( Q \). This gives us:

• \( x: -3 - 7 = -10 \)
• \( y: 0 - (-2) = 2 \)
• \( z: 6 - 6 = 0 \)

Thus, the direction vector is \( \vec{d} = (-10, 2, 0) \).

It tells us the line's overall direction, which we use to find further equations that describe the line itself.

Once we know our direction vector, we can move on to creating parametric equations. These equations break down the components of a line into individual expressions that depend on a parameter, usually \( t \). Parametric equations provide a way to express the line's coordinates in a controlled manner as \( t \) varies.

From the earlier found vector equation \( \vec{r}(t) = (7, -2, 6) + t(-10, 2, 0) \), we identify the parametric equations:

• \( x = 7 - 10t \)
• \( y = -2 + 2t \)
• \( z = 6 \)

Parametric equations are fantastic because they allow us to "walk" along the line. By plugging in different values for \( t \), we find different points on the line. The parameter \( t \) serves as a slider that moves us from one point to another along the line.

Symmetric equations offer another way to represent the same line. They eliminate the parameter \( t \), providing a format that directly agrees with each coordinate relation without the need for a parameter. This is useful when you want to see how coordinates relate directly to one another.

For the line defined by parametric equations \( x = 7 - 10t \) and \( y = -2 + 2t \), solving for \( t \) and equating the expressions results in:

• \( t = \frac{x - 7}{-10} \)
• \( t = \frac{y + 2}{2} \)

This leads to the symmetric equation:\[ \frac{x - 7}{-10} = \frac{y + 2}{2} \]Since \( z \) remains constant as \( z = 6 \), it joins our equation set without alteration. Symmetric equations give us a consistent format to view the line relation without delving into parameterized values.