91Ó°ÊÓ

In geometry, a normal vector is a vector that is perpendicular to a surface or a plane. In the context of a plane given by an equation like the one in our problem, the coefficients of the variables in the equation form the components of the normal vector.
For example:

• The equation of the plane is given by \(x - y + z = 8\) has a normal vector \(\mathbf{n_1} = \langle 1, -1, 1 \rangle\).
• Similarly, \(2x + y - z = -1\) provides the normal vector \(\mathbf{n_2} = \langle 2, 1, -1 \rangle\).

The significance of normal vectors is immense as they assist in determining relationships between planes, such as being parallel or perpendicular.

The dot product, also known as the scalar product, is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This operation plays a pivotal role in determining the angle between two vectors.
To calculate the dot product of two vectors, such as \(\mathbf{n_1} = \langle 1, -1, 1 \rangle\) and \(\mathbf{n_2} = \langle 2, 1, -1 \rangle\), you multiply their corresponding components and sum up the results:

• \(\mathbf{n_1} \cdot \mathbf{n_2} = 1 \times 2 + (-1) \times 1 + 1 \times (-1) = 2 - 1 - 1 = 0\)

When the dot product is zero, the vectors are perpendicular, indicating that their angle is 90 degrees. This makes the comparison beneficial for analyzing the orientation of planes in space.

Orthogonal vectors are an important concept, especially in higher dimensions and vector geometry. Two vectors are considered orthogonal if their dot product is zero. This shows that they are perpendicular, or at right angles to each other.
In the context of normal vectors from plane equations, \(\mathbf{n_1} = \langle 1, -1, 1 \rangle\) and \(\mathbf{n_2} = \langle 2, 1, -1 \rangle\) being orthogonal means that their respective planes are perpendicular to one another.
This concept is very powerful in space and plane geometry because recognizing orthogonal vectors can simplify complex physical interpretations and calculations in many applications, such as physics, engineering, and computer graphics.