91Ó°ÊÓ

The direction vector is a fundamental concept in understanding the orientation and trajectory of a line in space. When you are working with parametric equations of lines, the direction vector helps to define in which direction the line extends. For a line passing through two points, the direction vector can be found by subtracting the coordinates of one point from the other. For example, given two points, \(P_{1}(-3,1,-1)\) and \(P_{2}(7,11,-8)\), the direction vector \(\mathbf{d}\) is determined as follows:

• Subtract the x-coordinates: \(7 - (-3) = 10\)
• Subtract the y-coordinates: \(11 - 1 = 10\)
• Subtract the z-coordinates: \(-8 - (-1) = -7\)

Thus, the direction vector is represented as \langle 10, 10, -7\rangle\. This vector tells us how much we move in each coordinate direction per unit increase in the parameter \(t\).

The intersection of a line with a plane is where the line "cuts through" the plane. In three dimensions, we often explore intersections with the coordinate planes: XY-plane, XZ-plane, and YZ-plane. Each of these planes corresponds to setting one of the coordinates to zero. To find where a line intersects these planes, parametric equations can be used to substitute and solve:

• **XY-plane intersection:** Set \(z = 0\). Solve \(-1 - 7t = 0\) to find \(t = -\frac{1}{7}\). Insert this \(t\) in x and y parametric equations to locate the intersection.
• **XZ-plane intersection:** Set \(y = 0\). Solve \(1 + 10t = 0\) to find \(t = -\frac{1}{10}\). Use this in equations for \(x\) and \(z\) to determine the intersection point.
• **YZ-plane intersection:** Set \(x = 0\). Solve \(-3 + 10t = 0\) for \(t = \frac{3}{10}\). Substitute back into y and z equations to find this intersection.

Each solution gives the specific point where the line intersects each coordinate plane.

Coordinate geometry, or analytic geometry, is the study of geometry using a coordinate system. This allows us to represent geometric figures using numbers and algebra. In three-dimensional space, points are represented as \( (x, y, z) \), which gives each point a unique position based on three numbers.This concept is powerful:

• Enables easy calculation of geometric properties like distance and angles.
• Allows visualization of how shapes, lines, and planes interact.
• Facilitates finding intersections and relationships between geometric entities.

For instance, using coordinate geometry, you can efficiently calculate the point of intersection between a line and a plane by setting up and solving equations addressed earlier.

The parametric form of equations is a dynamic way to express lines using a parameter \(t\). Instead of expressing the line as a static equation, parametric equations describe the position of a point on the line based on the value of \(t\). The general format of a parametric equation for a line through a point \(P_1(x_1, y_1, z_1)\) with a direction vector \langle a, b, c \rangle\ is given by:

• \( x = x_1 + at \)
• \( y = y_1 + bt \)
• \( z = z_1 + ct \)

Using the points \(P_1(-3,1,-1)\) and direction vector \langle 10, 10, -7 \rangle\, the parametric equations become:

• \( x = -3 + 10t \)
• \( y = 1 + 10t \)
• \( z = -1 - 7t \)

This approach provides flexibility; by adjusting \(t\), you can obtain any point on the line.