91Ó°ÊÓ

In geometry, expressing a line in vector form makes visualizing and solving problems easier. When you see a line in vector form, it's defined using positional and directional components.
The general form is \[\mathbf{r} = \mathbf{a} + t\mathbf{b}\]

• \(\mathbf{r}\) is the position vector of any point on the line.
• \(\mathbf{a}\) is the position vector of a specific point on the line, usually given as a constant vector.
• \(t\) is a scalar parameter that changes values along the line.
• \(\mathbf{b}\) is the direction vector for the line, showing the direction and orientation.

For example, Line 1 in the exercise is expressed as \[\begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} -1 \ 3 \ 1 \end{bmatrix} + t \begin{bmatrix} 4 \ 1 \ 0 \end{bmatrix}.\]This means the line passes through the point \((-1, 3, 1)\) and extends in the direction determined by the vector \((4, 1, 0)\).
Understanding this form is crucial when solving problems related to lines in three-dimensional space.

Parametric equations are another way to represent lines, where each coordinate is expressed in terms of a single parameter. This parameter, often denoted as \( t \), allows us to find different points along the line. For instance, for Line 1 in the problem, we have:

• \(x = -1 + 4t\)
• \(y = 3 + t\)
• \(z = 1\).

These equations tell you how the coordinates change as \( t \) varies. The parameter \( t \) can be any real number. As \( t \) increases or decreases, it describes each point along the direction of the line.
Parametric equations are especially useful in solving problems where you need to find intersections or determine geometric properties influenced by different variables.

A plane in three dimensions can be described using what's called the plane equation. This is usually in the form:\[Ax + By + Cz = D\]where \(A, B,\) and \(C\) are coefficients that define the normal vector to the plane, and \(D\) is a constant.
In the exercise, once the intersection of the lines is determined, a plane is formed containing both lines. The normal vector to this plane is derived from the cross product of the direction vectors of the two lines:
\[\mathbf{d}_1 \times \mathbf{d}_2 = \begin{bmatrix} 3 \ -12 \ 12 \end{bmatrix}.\]The resulting plane equation simplified to:\[x - 4y + 4z = -25\]This formulation helps in various applications such as determining if a point lies on the plane or finding angles between planes.

Direction vectors play a crucial role in defining lines in space. They provide a sense of orientation and indicate where the line extends. Given in the form of a vector, it is usually seen as part of the vector equations for lines.
For Line 1 and Line 2 in the problem, the direction vectors were \(\mathbf{d}_1 = \begin{bmatrix} 4 \ 1 \ 0 \end{bmatrix}\) and \(\mathbf{d}_2 = \begin{bmatrix} 12 \ 6 \ 3 \end{bmatrix}\), respectively.
Understanding these vectors is essential because they:

• Indicate the line's orientation and path.
• Work as core components in calculating cross products when determining planes.
• Provide a method for calculating line intersections when combined with parameterization.

Thus, having a firm grasp of direction vectors enhances your ability to handle spatial reasoning and solve geometric problems efficiently.