Question

Find a parametric description of the line segment from the point \(P\) to the point \(Q\). The solution is not unique. $$P(-1,-3), Q(6,-16)$$

Short answer

Based on the given solution, create a short answer question: Question: Given points P(-1, -3) and Q(6, -16), find the parametric equations for the line segment between P and Q. Answer: The parametric equations for the line segment between points P(-1, -3) and Q(6, -16) are x = -1 + 7t and y = -3 - 13t.

Step-by-step solution

Find the vector between P and Q

First, we need to find the vector between points \(P\) and \(Q\). We can do this by subtracting the coordinates of point \(P\) from the coordinates of point \(Q\), and this will give us the displacement vector \(\overrightarrow{PQ}\). $$\overrightarrow{PQ} = Q - P = (6 - (-1), -16 - (-3)) $$ $$\overrightarrow{PQ} = (6 +1, -16+3) = (7, -13) $$ So, the vector \(\overrightarrow{PQ} = (7, -13)\).

Create a parametric equation along the line segment

Now that we have the vector \(\overrightarrow{PQ}\), we can use it to create a parametric equation along the line segment. We will use the point \(P\) as a starting point and scale the vector along the line segment by a parameter, \(t\). So, the parametric equation will be in the form: $$(x, y) = P + t\cdot\overrightarrow{PQ}$$ Replacing \(P\) and \(\overrightarrow{PQ}\) with their coordinate values, we have: $$(x, y) = (-1, -3) + t \cdot (7, -13)$$

Separate parametric equations for x and y

Now, we can write separate parametric equations for \(x\) and \(y\): $$x = -1 + 7t$$ $$y = -3 - 13t$$ These are the parametric equations for the line segment between points \(P\) and \(Q\). Note that the solution is not unique, meaning other forms of the parametric equation may also correctly describe the line segment.

Key concepts

Vector Subtraction

When working with coordinate geometry, vector subtraction is a crucial concept. Imagine we have two points, point \(P\) and point \(Q\), where \(P(-1, -3)\) and \(Q(6, -16)\). To find the vector from one point to another, we subtract coordinates of the starting point from the coordinates of the ending point.
This subtraction gives us the displacement vector that tells us the direction and distance from one point to the other.

• For the x-coordinate: Subtract \(-1\) from \(6\), resulting in \(7\).
• For the y-coordinate: Subtract \(-3\) from \(-16\), resulting in -13.

The result \((7, -13)\) indicates that moving from \(P\) to \(Q\) involves a shift of 7 units to the right and 13 units down. This operation establishes the displacement vector \(\overrightarrow{PQ}\).

Line Segment

One might ask, how do we represent a line segment joining two points in a coordinate plane? A line segment is simply the shortest path connecting two endpoints in geometry. When describing the segment from \(P(-1, -3)\) to \(Q(6, -16)\), it is crucial to understand how to express it while maintaining the direction along this path.
Parametrically, it can be expressed as a function relating a position on the line based on a parameter \(t\). It moves from \(t=0\) (at point \(P\)) to \(t=1\) (at point \(Q\)).

• The use of \(t\) enables us to scale and travel along the line.
• This method helps show every point between \(P\) and \(Q\).

Thus by setting a parameter, combining initial point \((P)\) with the displacement vector \(\overrightarrow{PQ}\), results in a parametric equation for the line segment.

Displacement Vector

The displacement vector is a powerful concept in coordinate geometry. This vector delineates the direction and magnitude of movement from one point to another. For points \(P(-1, -3)\) and \(Q(6, -16)\), we found \(\overrightarrow{PQ} = (7, -13)\).
This is the basis for forming the parametric equation.

• The displacement vector's components \((7, -13)\) reveal how far and in what direction the destination point \(Q\) is from the starting point \(P\).
• When constructing parametric equations, this vector is scaled by a parameter \(t\), indicating movement along the direction of the vector.

This approach not only spans the direct path but also can parameterize paths in physics or engineering, shown similarly through formulas.

Coordinate Geometry

In coordinate geometry, points are defined using pairs of numbers, and geometric figures can be described using algebraic equations. This branch is crucial for understanding locations and constructing figures on a plane.
In our case, given points \(P\) and \(Q\), coordinate geometry allows us to determine their relative position.

• With points defined as \((x, y)\), we use these components to apply operations like vector subtraction.
• Through these operations, including finding displacement vectors and creating parametric equations, we can model elements such as line segments and trajectories.

Parametric equations in coordinate geometry simplify expressing movement where traditional equations fall short. This method provides clarity when defining directional paths between points on the plane.