Question

Find a parametric description of the line segment from the point \(P\) to the point \(Q\). The solution is not unique. $$P(1,3), Q(-2,6)$$

Short answer

The parametric description of the line segment connecting points P(1, 3) and Q(-2, 6) is (x, y) = (1 - 3t, 3 + 3t), with parameter t varying between 0 and 1.

Step-by-step solution

Calculate the direction vector

First, we need to find the direction vector connecting points P and Q. The direction vector can be found by taking the difference between the coordinates of point Q and point P. $$\vec{v} = Q - P = (-2-1, 6-3) = (-3,3)$$ The direction vector is \(\vec{v} = (-3,3)\).

Define the parametric equation

Now that we have the direction vector, we can define the parametric equation of the line segment. The equation will be in the form: $$(x,y) = P + t\cdot\vec{v}$$ where \((x,y)\) represent the coordinates on the line segment, \(P\) is the initial point, \(\vec{v}\) is the direction vector, and \(t\) is a parameter that varies from 0 to 1 (0 corresponds to point P and 1 corresponds to point Q).

Substitute and simplify the equation

Inserting the values for \(P\) and \(\vec{v}\) from Step 1: $$(x,y) = (1,3) + t\cdot(-3,3)$$ Now simplify the equation: $$(x,y) = (1-3t, 3+3t)$$ The parametric description of the line segment is \((x,y) = (1-3t, 3+3t)\) with parameter \(t\) varying between 0 and 1.

Key concepts

Direction Vector

Understanding the direction vector is crucial in parametric equations. It tells us the direction in which a line travels.
This vector is calculated by subtracting corresponding coordinates of two points on the line. In context, if we have two points, say, \(P(x_1, y_1)\) and \(Q(x_2, y_2)\), then the direction vector \(\vec{v}\) is given by:

• \(\vec{v} = (x_2 - x_1, y_2 - y_1)\)

In our example, with points \(P(1, 3)\) and \(Q(-2, 6)\), we calculate:

• \(\vec{v} = (-2 - 1, 6 - 3) = (-3, 3)\)

The vector \((-3, 3)\) indicates that starting from \(P\), we move left 3 units and up 3 units to reach \(Q\).
This consistent direction helps in defining the path between these points, and is essential for the parametric equation.

Line Segment

A line segment is a specific portion of a line that connects two points. It does not extend indefinitely like a line but has definitive endpoints.
In parametric terms, we describe this line segment by using a parameter \(t\) that varies from 0 to 1, representing different points on the segment.
When \(t = 0\), the point on the line segment is exactly \(P\), and when \(t = 1\), the point on the line segment is exactly \(Q\).The line segment from \(P\) to \(Q\) can be expressed as:

• \((x, y) = P + t \cdot \vec{v}\)

From this equation:

• At \(t = 0\), \((x, y) = P\),
• At \(t = 1\), \((x, y) = Q\).

This helps in visualizing any point on the line segment relative to its endpoints.
It is a powerful way to understand linear interpolation between two points in geometry.

Points in Geometry

Points are fundamental objects in geometry and serve as precise locations in the plane. When describing lines, segments, or shapes, points act as key reference markers.
Each point is defined by coordinates on a plane, like \((x, y)\), with \(x\) being the horizontal position and \(y\) being the vertical position.In our case, we have two points, \(P(1, 3)\) and \(Q(-2, 6)\):

• Point \(P\) has an \(x\)-coordinate of 1 and a \(y\)-coordinate of 3.
• Point \(Q\) has an \(x\)-coordinate of -2 and a \(y\)-coordinate of 6.

These coordinates help us understand the position of these points in the plane and form the basis for calculating distances, directions, and creating parametric equations.
Points can also be used to define shapes, such as triangles or rectangles, adding depth to our understanding of geometry.