Question

Find a parametric description of the line segment from the point \(P\) to the point \(Q\). The solution is not unique. $$P(-8,2), Q(1,2)$$

Short answer

Question: Given the points P(-8, 2) and Q(1, 2), find the parametric equation of the line segment connecting these points. Answer: The parametric equation of the line segment connecting points P(-8, 2) and Q(1, 2) is given by the system: $$\begin{cases} x = -8 + 9t \\ y = 2 \end{cases}$$

Step-by-step solution

Finding the vector PQ

Let's find the vector connecting points P and Q. To do this, simply subtract the coordinates of point P from the coordinates of point Q: $$\vec{PQ} = Q - P = (1 - (-8), 2 - 2) = (9, 0)$$

Writing the parametric equation

Now we can write the parametric equation of the line segment from P to Q. We will use a parameter t that varies between 0 and 1: $$\vec{PQ} = \vec{OP} + t(\vec{OQ} - \vec{OP})$$ Where \(\vec{OP}\) and \(\vec{OQ}\) are the position vectors of points P and Q with respect to the origin O, and t is the parameter. First, let's find the position vectors of points P and Q: $$\vec{OP} = -8\hat{i} + 2\hat{j}$$ $$\vec{OQ} = 1\hat{i} + 2\hat{j}$$ Now, substitute these position vectors and the vector PQ we found earlier in the parametric equation: $$\vec{PQ} = (-8\hat{i} + 2\hat{j}) + t((1\hat{i} + 2\hat{j}) - (-8\hat{i} + 2\hat{j}))$$ $$\vec{PQ} = (-8\hat{i} + 2\hat{j}) + t(9\hat{i} + 0\hat{j})$$ Now, find the parametric equation of the line segment from P to Q by separating the i and j components: $$x = -8 + 9t$$ $$y = 2$$ Our final parametric equation of the line segment is: $$\begin{cases} x = -8 + 9t \\ y = 2 \end{cases}$$

Key concepts

Vector Operations

Vector operations are fundamental tools in mathematics, especially in geometry and physics. Vectors are used to describe quantities that have both a magnitude and a direction, such as velocity, force, and displacement. When working with vectors, you often perform operations like additions, subtractions, dot products, and cross products.

• Addition/Subtraction: You add or subtract vectors by combining their corresponding components. For example, if \(\vec{a} = (a_1, a_2)\) and \(\vec{b} = (b_1, b_2)\), then \(\vec{a} + \vec{b} = (a_1 + b_1, a_2 + b_2)\).
• Scalar Multiplication: Multiplying a vector by a scalar involves multiplying each component by that scalar, which stretches or shrinks the vector without changing its direction unless the scalar is negative.
• Magnitude: The length of a vector can be found using the Pythagorean theorem, \(|\vec{a}| = \sqrt{a_1^2 + a_2^2}\).

In the given exercise, we used vector subtraction to find the vector \(\vec{PQ}\) connecting two points \(P(-8,2)\) and \(Q(1,2)\). This operation gives us the direction from \(P\) to \(Q\) as well as the length of the line segment between them. Understanding these operations sets us up for tackling more complex problems like writing parametric equations.

Line Segment in Space

A line segment in space is a part of a line that is bounded by two endpoints. It includes all the points on the line between these endpoints. In two dimensions, like in this exercise, a line segment is flat. However, the concept can be extended to three dimensions, where a line segment can pass through planes and space.
To describe a line segment mathematically, parametric equations are often used. These equations relate each coordinate (like \(x\), \(y\), and sometimes \(z\)) to a parameter \(t\) that varies over a specific interval. In our example, \(t\) ranges from 0 to 1, where \(t = 0\) corresponds to the start of the segment, and \(t = 1\) corresponds to the end.

• The parametric form of the line segment from \(P\) to \(Q\) was found to be \(x = -8 + 9t\) and \(y = 2\). This shows that as \(t\) goes from 0 to 1, the \(x\) coordinate linearly increases from \(-8\) to \(1\), while \(y\) remains constant at \(2\).
• Parametric equations are beneficial because they not only provide the direction and length but also allow us to easily handle geometric transformations and movements in space.

Two-Dimensional Geometry

Understanding two-dimensional geometry is crucial for solving problems involving shapes, lines, and angles on a plane. It involves the study of figures like points, lines, and polygons, and uses various techniques and formulae to understand properties and relationships between them.
In two-dimensional geometry, lines can be described by different forms of equations like slope-intercept form, standard form, or parametric form as seen in our exercise.

• Points: A point is defined by its coordinates, for example, \(P(-8,2)\) and \(Q(1,2)\), using ordered pairs.
• Lines: A line can be described by an equation such as \(y = mx + b\) or parametrically, and a straight path connecting two points.
• Distance: The distance between two points can be derived using the distance formula, but in this exercise, the vector connecting the two points helps us understand the line segment directly.

The concepts learned in two-dimensional geometry are essential as they build a foundation for more advanced studies, including calculus and physics. This forms a basis for understanding how shapes and spaces work, whether they are lines, curves, or two-dimensional surfaces.