91Ó°ÊÓ

Vector algebra is a branch of mathematics that deals with quantities having both magnitude and direction. In the context of planes and lines in 3D space, vectors help us determine directions and relationships between lines and planes.
To interpret a vector, you think of it as an arrow that points from one location to another. The direction vector, such as \( \mathbf{d} = (3, 2, 4) \), provides the direction in which our line extends.
When two lines share the same direction vector, as in our exercise, it indicates that they are parallel. Knowing two vectors on a plane, such as our direction vector and the vector \( \mathbf{v} = (-4, 5, -5) \) between two points, helps us to define that plane, provided these vectors are linearly independent.

• Linear independence means the vectors are not scalar multiples of each other, allowing them to span a plane.
• If two vectors are dependent (one is a multiple of the other), they do not define a plane as they lie along the same line.

The parametric form of a plane is a powerful concept in mathematics, useful for describing a plane using two parameters, allowing any point on the plane to be expressed. It provides flexibility in representing all points on a plane in 3D geometry.
In our solution, the parametric form is given by \( \mathbf{r}(s, t) = (1, -1, 5) + s(3, 2, 4) + t(-4, 5, -5) \). This equation describes all positions \( \mathbf{r} \) on the plane as a combination of a fixed point and scaled vectors.
The vectors \( (3, 2, 4) \) and \( (-4, 5, -5) \) are direction vectors achieving any point on the plane when adjusted by changing parameters \( s \) and \( t \).

• Start from the fixed point \( (1, -1, 5) \), which lies on the plane.
• Adjust \( s \) and \( t \) to navigate across the plane in the direction of each vector.
• Each combination of \( s \) and \( t \) leads to a unique point on the plane, showcasing the plane's entire surface.

The Cartesian form of a plane equation is a familiar way to express a plane in mathematics using algebraic terms. It is straightforward and uses a normal vector to define the plane's orientation in space.
In the exercise, the Cartesian equation is found as \( 3x + 2y + 4z = 27 \). This form relates the variables \( x, y, \) and \( z \) with the plane's position by representing how far each direction reaches.
To define this form:

• The coefficients \( 3, 2, \) and \( 4 \) emerge from the normal vector \( (3, 2, 4) \), establishing the plane's orientation.
• Point substitution, such as substituting \((1, -1, 5)\), helps derive the constant on the right, \( 27 \), ensuring the equation represents the desired plane through the chosen point.

Forming a plane using Cartesian coordinates is effective for solving many practical problems, offering an algebraic way to grasp a plane's spatial properties.