Problem 25 - Thomas Calculus

Question

Find a parametrization for the curve. the ray (half line) with initial point \((2,3)\) that passes through the point \((-1,-1)\)

Step-by-Step Solution

Answer

The parametrization is \( x(t) = 2 - 3t \), \( y(t) = 3 - 4t \) for \( t \geq 0 \).

Step 1: Understand the Ray's Direction

To find the direction of the ray, subtract the coordinates of the initial point from the point through which the ray passes. This gives us the direction vector.The direction vector \( \mathbf{v} \) is calculated as:\( \mathbf{v} = (-1 - 2, -1 - 3) = (-3, -4) \).

Step 2: Define the Parametrization Variables

The equation of the ray can be represented by the equation of a line in parametric form. We need a parameter \( t \) that will describe the positions along the ray. The parameter \( t \) should start from zero, corresponding to the initial point, and increase in the direction of the vector.

Step 3: Write the Parametric Equations

Use the initial point and the direction vector to write the parametric equations:The point on the curve at any parameter value \( t \) can be given by:\[x(t) = 2 + (-3)t = 2 - 3t\]\[ y(t) = 3 + (-4)t = 3 - 4t\]

Step 4: Specify Parameter Constraints

Since we are dealing with a ray starting at \((2,3)\) and extending through \((-1,-1)\), the parameter \( t \) must be non-negative, i.e., \( t \geq 0 \). This restriction ensures we only describe the ray and not the entire line.

Key Concepts

Direction Vector

When dealing with parametrization of curves, understanding the direction vector is crucial. A direction vector indicates the direction in which a curve, like a ray or line, extends. In this case, the direction vector is derived by subtracting the coordinates of the initial point from those of another point along the ray.

For the given exercise, the ray starts at point \(2,3\) and passes through \(-1,-1\). To find its direction vector, we subtract the initial point from the passing point:

• Subtract the x-coordinates: \(-1 - 2 = -3\).
• Subtract the y-coordinates: \(-1 - 3 = -4\).

Thus, the direction vector \(\mathbf{v}\) is \((-3, -4)\). This vector defines the path on which the ray travels, giving it both direction and magnitude.

The direction vector is essential as it allows us to understand how the curve stretches in space and helps us set up the correct equations to describe that motion.

Parametric Equations

Parametric equations provide a way to describe a curve using a parameter. In the case of a ray or line, this parameter is often denoted as \(t\). The purpose of using parametric equations is to express both the x and y coordinates as functions of this parameter, enabling easier manipulation and analysis of the curve.

To set up parametric equations for the ray beginning at \(2,3\) with direction vector \((-3, -4)\), we start at the initial point and add scaled portions of the direction vector. This translates into the equations:

• For the x-coordinate: \[ x(t) = 2 - 3t \]
• For the y-coordinate: \[ y(t) = 3 - 4t \]

Here, \(t\) acts as our slider. When \(t = 0\), the position described is exactly the starting point \(2,3\). As \(t\) increases, the point \(x(t), y(t)\) moves along the ray in the direction given by our direction vector. This format makes it simple to track how the curve progresses along its specified path.

Parameter Constraints

When defining a curve using parametric equations, specifying parameter constraints is very important. These constraints dictate the beginning and end of the curve's path. In the context of a ray, which is essentially a half-line, it's vital to limit the parameter so it describes only the portion of the line wanted.

In our example, the ray originates at point \(2,3\) and extends through \(-1,-1\). We choose the parameter \(t\) to be non-negative \(t \geq 0\), which means the ray extends only in one direction from the point \(2,3\).

• This lower bound ensures we don't describe a full line, which would extend infinitely in both directions from any point.
• For any \(t < 0\), we would describe points that lie on the opposite extension of the line, which is not part of our ray.

Constrained in this way, the parametric equations capture exactly the portion of the line needed, saving us from inadvertently describing a line segment or a line stretching infinitely in both directions.