Question

Mutually Orthogonal In your own words, describe the relationship between two families of curves that are mutually orthogonal.

Short answer

Two families of curves are mutually orthogonal if every curve in the first family intersects every curve in the second family at right angles. In other words, they meet each other perpendicularly or orthogonally at all points of intersection.

Step-by-step solution

Definition of Orthogonal

Orthogonal in the geometric sense means 'right-angled'. It is a term used to describe the perpendicularity of lines or curves. If two lines or curves are orthogonal, they meet at a right angle, i.e., an angle of 90 degrees.

Description of Families of Curves

Now, a family of curves is a set of related curves described by the same mathematical equation. Each curve represents a different value or set of values for the parameters of the equation.

Mutually Orthogonal Families of Curves

Let's apply our definitions. Two families of curves are mutually orthogonal if every curve in the first family intersects every curve in the second family at right angles. This relationship is maintained at all points of intersection between curves from the two families.

Example of Mutually Orthogonal Families of Curves

Consider the families of lines with equations in the form \(y = mx\) and \(y = -x/m\). For any choice of the parameter m (which is the slope of the line), lines from the first family are perpendicular to lines from the second family. Here, these two families of lines are mutually orthogonal.

Key concepts

Orthogonal Intersection

In geometry, the term "orthogonal intersection" refers to the specific instance where two lines or curves meet at right angles. Imagine penciling two lines on a piece of paper so that they intersect in the shape of a "T" or an "+"; this is what we mean by orthogonal.

An orthogonal intersection is a fundamental concept because it reveals a natural balance and symmetry in geometric shapes. It intersects at a 90-degree angle, creating a perfect perpendicular meeting point:

• This principle holds not only for single lines but also when we consider families of curves.
• An orthogonal intersection between two families of curves means that curves from one family consistently intersect the curves from another family at these right angles.

Thus, whenever you encounter the term "orthogonal," think of those perfect right angles that simplify complex geometric relationships into a clear understanding.

Perpendicular Lines

Understanding perpendicular lines is intrinsic to mastering the concept of orthogonality. Perpendicular lines are two lines that intersect in such a way that they divide the plane into four equal angles of 90 degrees each.

In essence:

• These lines create a cross shape, forming squares at the point of intersection.
• The perfect 90-degree angles are a hallmark of perpendicularity, indicative of balance and harmony.

This perpendicular nature ensures that the product of their slopes is \(-1\). Thus, any line of slope \(m\) that is perpendicular to another will have a slope of \(-1/m\). This relationship is crucial in identifying perpendicular lines in algebraic contexts. For example, the lines described by equations \(y = mx\) and \(y = -x/m\) are perpendicular by this definition.

Families of Curves

Whenever we discuss families of curves, we refer to a collection of curves represented by the variation in one or more parameters of a specific equation.

For example, consider the equation of a line \(y = mx\). The family here includes all the lines where the parameter \(m\) or slope varies:

• This implies an infinite set of lines, each defined by its unique slope \(m\).
• Each line in this family still maintains the same y-intercept, essential in keeping them "related."

In the context of mutually orthogonal families, each curve from one family intersects every curve from another family orthogonally. It creates a harmonious geometric dance, structured yet diverse. Understanding this connection allows mathematicians to explore complex systems where symmetry and balance are crucial between intersecting planes or space.