91Ó°ÊÓ

When you talk about planes in three-dimensional geometry, it’s crucial to understand the role of **normal vectors**. A normal vector is simply a vector that is perpendicular to a plane. It provides important information about the plane's orientation within the space.

Every plane surface can be represented mathematically by an equation of the form \(ax + by + cz = d\), where \(a\), \(b\), and \(c\) are constants. These constants play a significant role because they determine the normal vector of the plane. Specifically, the normal vector is \((a, b, c)\).

Let's consider an example: the plane described by the equation \(x + y + 4z = 10\). The normal vector for this plane is \((1, 1, 4)\). Similarly, for the plane \(-x - 3y + z = 10\), the normal vector is \((-1, -3, 1)\).

Normal vectors help to compare planes by determining how they relate to each other. They are key in understanding concepts like parallelism and orthogonality between planes.

The **dot product** is a mathematical operation that takes two equal-length sequences of numbers (often vectors) and returns a single number. It’s a fundamental tool for vector calculations, especially useful in determining angles between two vectors.

To calculate the dot product of two vectors \(\mathbf{a} = (a_1, a_2, a_3)\) and \(\mathbf{b} = (b_1, b_2, b_3)\), you multiply their corresponding components and then sum up these products:

\[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \]

The dot product is extremely helpful in determining whether two vectors are perpendicular to each other. For example, when finding if planes are orthogonal, like in our problem, you calculate the dot product of their normal vectors.

If the dot product equals zero, the vectors are perpendicular, indicating the planes are orthogonal. In our case, the normal vectors \((1, 1, 4)\) and \((-1, -3, 1)\) were found to have a dot product of 0:

\[(1)(-1) + (1)(-3) + (4)(1) = -1 - 3 + 4 = 0\]

This result confirmed that the planes are orthogonal.

Understanding **vector perpendicularity** is key to grasping the relationship between geometric entities such as planes. When two vectors are perpendicular, it implies that they meet at a right angle. In the context of planes, this concept helps in determining if two planes are orthogonal.

For vectors \(\mathbf{u}\) and \(\mathbf{v}\) to be perpendicular, their dot product must be zero:

\[ \mathbf{u} \cdot \mathbf{v} = 0 \]

In the problem we considered, we checked for perpendicularity between normal vectors \((1, 1, 4)\) and \((-1, -3, 1)\). By computing their dot product, we found it to be zero, thus confirming their perpendicular nature.

Perpendicular vectors in three-dimensional space indicate that they span two individual directions that form no overlap, reflecting a clear 90-degree directional relationship. This property is used to verify that the two planes in question meet at a right angle, or are orthogonal. Understanding this principle is essential in solving geometry-related problems where vector relationships dictate the properties of more complex shapes.