Question

Orthogonal lines Recall that two lines \(y=m x+b\) and \(y=n x+c\) are orthogonal provided \(m n=-1\) (the slopes are negative reciprocals of each other). Prove that the condition \(m n=-1\) is equivalent to the orthogonality condition \(\mathbf{u} \cdot \mathbf{v}=0,\) where \(\mathbf{u}\) points in the direction of one line and \(\mathbf{v}\) points in the direction of the other line..

Short answer

Question: Prove that the condition \(mn=-1\) is equivalent to the orthogonality condition \(\mathbf{u} \cdot \mathbf{v}=0\) for two given lines with slopes \(m\) and \(n\). Short Answer: By converting the lines into parametric form, finding their direction vectors, and calculating their dot product, we can show that if \(mn=-1\), the dot product of their direction vectors is zero, which proves orthogonality. Conversely, if the dot product is zero, the condition \(mn=-1\) holds, demonstrating their equivalence.

Step-by-step solution

Convert the equations of both lines into parametric form

To find the direction vectors of the given lines, we must first write them in parametric form. Let's convert the given lines \(y=mx+b\) and \(y=nx+c\) into parametric form by introducing a parameter \(t\). For the first line, \(y=mx+b\): $$ x = t \\ y = mt + b $$ For the second line, \(y=nx+c\): $$ x = s \\ y = ns + c $$

Find the direction vectors of both lines

Now, from the parametric form of both lines, we find the direction vectors \(\mathbf{u}\) and \(\mathbf{v}\). For the first line, \(\mathbf{u} = \begin{pmatrix} 1 \\ m \end{pmatrix}\). For the second line, \(\mathbf{v} = \begin{pmatrix} 1 \\ n \end{pmatrix}\).

Calculate the dot product of \(\mathbf{u}\) and \(\mathbf{v}\)

The dot product between the two direction vectors is given by: $$ \mathbf{u} \cdot \mathbf{v} = \begin{pmatrix} 1 \\ m \end{pmatrix} \cdot \begin{pmatrix} 1 \\ n \end{pmatrix} = 1\cdot 1 + m\cdot n = 1 + mn $$

Prove orthogonality condition and slopes condition are equivalent

To show that the condition \(mn=-1\) is equivalent to the orthogonality condition \(\mathbf{u} \cdot \mathbf{v}=0\), we simply substitute \(mn=-1\) into our equation for the dot product of \(\mathbf{u}\) and \(\mathbf{v}\): $$ 1 + mn = 1 + (-1) = 0. $$ Thus, if the slopes of the given lines satisfy \(mn=-1\), then the dot product of their direction vectors is zero, proving the orthogonality condition is satisfied. Conversely, if the orthogonality condition holds, that is \(\mathbf{u} \cdot \mathbf{v} = 0\), then from the dot product equation, we can write: $$ 1 + mn = 0 \\ mn=-1 $$ Hence, the condition \(mn=-1\) is equivalent to the orthogonality condition \(\mathbf{u} \cdot \mathbf{v}=0\).

Key concepts

Parametric Form of Lines

In geometry, understanding the parametric form of lines is crucial for representing them in a more flexible way than the traditional slope-intercept form. Parametric equations allow us to express the coordinates of points on a line in terms of a single variable, typically known as a parameter, which is often denoted by the letter 't' or 's'.

For a line with equation in slope-intercept form, such as \(y = mx + b\), we can express the x-coordinate as \(x = t\), where 't' ranges over all real numbers. Consequently, we can then describe the y-coordinate in terms of 't' by substituting 't' into the original equation to get \(y = mt + b\), thus creating a pair of parametric equations:

• \(x = t\)
• \(y = mt + b\)

This form is particularly useful when studying lines in higher dimensions, intersection points, and provides a foundation for more advanced topics such as vector equations and 3D modeling.

Direction Vectors

Direction vectors are the backbone of vector geometry and provide a way to codify the direction and steepness of lines in a compact form. They are essentially vectors parallel to the line that can be derived from the parametric form of the line. Each direction vector consists of components that correspond to the x and y increments as the parameter varies.

For example, the direction vector of a line in parametric form with equations \(x = t\) and \(y = mt + b\) would be represented by \(\mathbf{u} = \begin{pmatrix} 1 \ m \end{pmatrix}\), where '1' is the increment in x for a unit increase in 't', and 'm' is the corresponding increment in y. These vectors are the building blocks for more complex operations in vector algebra including calculating angles between lines and determining line intersections.

Dot Product

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 is significant in determining the angle between the two vectors and is defined by multiplying corresponding components of the two vectors and then adding those products.

Mathematically, the dot product of two vectors \(\mathbf{u}\) and \(\mathbf{v}\), each containing two components, is calculated as \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 \), where \(u_1\) and \(u_2\) are the components of vector \(\mathbf{u}\) and \(v_1\) and \(v_2\) are those of vector \(\mathbf{v}\). When the dot product equals zero, it implies that the vectors are orthogonal, meaning they meet at a right angle, an essential concept when dealing with orthogonality of lines.

Negative Reciprocal Slopes

When discussing the concept of orthogonality in the context of linear equations on a plane, slopes play a central role. Two non-vertical lines are orthogonal (perpendicular) if and only if the product of their slopes is -1. In such a case, the slopes are said to be negative reciprocals of each other.

In mathematical terms, if one line has a slope \(m\), and another line is orthogonal to it, the slope of this second line will be \(n\), such that \(mn = -1\). This means if line one is rising, the orthogonal line will be falling at a rate that when multiplied together, they cancel each other out to -1. This concept is invaluable in coordinate geometry for designing and understanding complex shapes, creating grid layouts, and solving problems related to angles and distances between lines.