91Ó°ÊÓ

When working with planes in three-dimensional space, the concept of normal vectors is essential. A normal vector is a vector that is perpendicular to a plane. Think of it as an arrow sticking straight out from the surface of the plane.
The equation of a plane in the form \(ax + by + cz = d\) will have a corresponding normal vector \(\mathbf{n} = (a, b, c)\). This vector is crucial for determining the orientation of the plane.
For example, given the equations of the planes:

• First plane: \(5x + y - z = 10\) has the normal vector \(\mathbf{n_1} = (5, 1, -1)\).
• Second plane: \(x - 2y + 3z = -1\) has the normal vector \(\mathbf{n_2} = (1, -2, 3)\).

These normal vectors are used to understand the spatial relationship between the planes.

The dot product is a mathematical tool used to assess the angle between two vectors.
It is calculated by multiplying the corresponding components of two vectors and summing those products.
The equation for the dot product of two vectors \(\mathbf{a} = (a_1, a_2, a_3)\) and \(\mathbf{b} = (b_1, b_2, b_3)\) is:

• \(\mathbf{a} \cdot \mathbf{b} = a_1 \cdot b_1 + a_2 \cdot b_2 + a_3 \cdot b_3\).

For the planes in our exercise:

• The dot product \(\mathbf{n_1} \cdot \mathbf{n_2}\) translates into \(5 \cdot 1 + 1 \cdot (-2) + (-1) \cdot 3 = 0\).

If the dot product is zero, the vectors are perpendicular, indicating that the planes are orthogonal.
This property is very useful in geometry, and especially in problems involving angles and distances.

Understanding the concept of orthogonal, or perpendicular, planes is important in geometry. In three-dimensional space, two planes are orthogonal when their normal vectors are perpendicular to each other.
This means that the dot product of their normal vectors is zero.
Orthogonal planes can be visualized as meeting at a right angle to each other, like the walls of a room.
In our specific problem, the dot product of the normal vectors \(\mathbf{n_1} = (5, 1, -1)\) and \(\mathbf{n_2} = (1, -2, 3)\) was found to be zero.
Therefore, this indicates that the given planes are indeed orthogonal. This is a geometric indicator that the planes meet at a 90-degree angle, making them perpendicular in space.