91Ó°ÊÓ

A direction vector describes the direction and magnitude of a line in space.
In the context of 3D coordinate geometry, the direction vector is determined by finding the difference between two points that lie on the line.
For the given points \(A(6, 5, -2)\) and \(B(5, 1, 2)\), the direction vector \(\mathbf{d}\) is found by subtracting the coordinates of point \(A\) from point \(B\).
This results in the vector \(\mathbf{d} = B - A = (5 - 6, 1 - 5, 2 - (-2)) = (-1, -4, 4)\).
This direction vector shows that for every one unit you move along the x-axis, you move four units down the y-axis and four units up the z-axis.
The calculated direction vector is then used in forming parametric equations for the line.

3D coordinate geometry helps us understand spatial locations and directions.
We use a three-dimensional Cartesian coordinate system where every point in space is defined by \(x\), \(y\), and \(z\) coordinates.
These axes intersect at the origin \(O(0,0,0)\) and are perpendicular to each other.
In this coordinate system, lines are defined by points and direction vectors.
For instance, given points \(A(6, 5, -2)\) and \(B(5, 1, 2)\), we can determine a unique line passing through these points with a distinct direction.
The points provide specific positions in space, while the direction vector shows the path you would follow to move from one point to another along the line.
This system allows for precise and clear descriptions of geometric shapes and lines in three-dimensional space.

The parametric form of a line in 3D uses a parameter \(t\) to define the coordinates of any point on the line.
Given a point \(P_0 (x_0, y_0, z_0) \) that the line passes through and a direction vector \( \mathbf{d} = (a, b, c) \), we can write the parametric equations as:
\[ x = x_0 + at \] \[ y = y_0 + bt \] \[z = z_0 + ct \]
Each parameter \(t\) value corresponds to a specific point on the line.
For our example, substituting \( P_0 (6, 5, -2) \) and \( \mathbf{d} = (-1, -4, 4) \):
\( x = 6 - t\)
\( y = 5 - 4t \)
\( z = -2 + 4t \)
This form translates three variable equations directly into a line that we can easily understand and visualize, opening new doors for analysis and applications in various fields.