91Ó°ÊÓ

A plane in three-dimensional space can be defined using a plane equation. This equation is derived from a point through which the plane passes and a normal vector that is perpendicular to the plane. Consider a point

• defined by coordinates egin{align*}(x_0, y_0, z_0),e.g.,(0, -2, 1)

and a normal vector

• defined bythe tripletegin{align*}(a, b, c),e.g.,(-1, 1, -1).

The plane equation is then expressed as:\[ a(x-x_0) + b(y-y_0) + c(z-z_0) = 0 \] . This represents all points egin{align*}(x, y, z),each located on the plane.
after expanding and simplifying, we get our plane equation as

• \[-x + y - z = -3.\]

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Orthogonality in vectors is a crucial concept, especially when relating lines and planes in geometry. Two vectors are orthogonal, or perpendicular, if their dot product is zero.
If path of a line is to be parallel to a plane, its direction vector must be orthogonal to the plane's normal vector.
The normal vector of a plane can guide us in defining this relation.

• Using this concept, the normal vector is \[\mathbf{n} = \left[ -1, 1, -1 \right]\],

• The direction vector for a line that is parallel to the plane should satisfy \[ \mathbf{n} \cdot \mathbf{d} = 0 \]

where in \[\mathbf{d}\] is – the direction vector of the line. Thisgives usthe equation\[-a + b - c = 0\],providing specific constraints to selectcertain vectors.
This makes it straightforward to generate a direction vector for use indescribing a line parallelto the plane.

Parametric equations offer a highly flexible way of describing the path of a line in space. Using a point and direction vector, the line can be expressed as a set of equations depending on a parameter, generally denoted as \( t \).

• Consider a point through which the line passes, e.g.,\((5, -1, 0)\),

• and a direction vector, such as\(\mathbf{d} = \left[1, 0, 1 \right]\).

We express the line using parametric equations:

• \[x = 5 + t,\]
• \[y = -1,\]
• \[z = t.\]

Here, \( t \)can be varied, allowing us to pinpoint any specific location along the line,

streamlining calculations and enhancing intuitive understanding of its geometry.

The direction vector is critical when determining the orientation and path of a line. This vector indicates which direction the line extends within a vector space. It plays a pivotal role in constructing parametric equations.

• To find a direction vector parallel to a given plane, we should ensure it is orthogonal to the plane's normal vector.
• In an example where a normal vector is \[ \mathbf{n} = \left[ -1, 1, -1 \right] \],

we choose a direction vector like

• \( \mathbf{d} = \left[ 1, 0, 1 \right] \),

satisfying:\[-1\cdot 1 + 1\cdot 0 - 1 \cdot 1 = 0.\]

• This selection guarantees that the line remains orthogonal and parallel to the plane's normal vector, ensuring trueparallelism.

Using this vector provides a backbone for further geometric scenarios or for translating restrictions into practical solutions.