Question

Which of the following parametric equations describe the same line? a. \(x=3+t, y=4-2 t ;-\infty

Short answer

Answer: Equations a and b describe the same line.

Step-by-step solution

Identify direction vectors and points for each equation

Let's rewrite each parametric equation in a better form to easily find the direction vectors and points. a. \(\begin{cases} x=3+t \\ y=4-2t \end{cases}\) b. \(\begin{cases} x=3+4t \\ y=4-8t \end{cases}\) c. \(\begin{cases} x=3+t^3 \\ y=4-t^3 \end{cases}\) For each equation, setting parameter t = 0, we will get the corresponding point on the line (it doesn't matter which point, just need one): a. \((3,4)\) b. \((3,4)\) c. \((3,4)\) For each parametric equation, find their corresponding direction vector by taking the derivatives with respect to t of both x and y: a. \(\begin{cases} \frac{dx}{dt}=1 \\ \frac{dy}{dt}=-2 \end{cases}\) -> direction vector \((1,-2)\) b. \(\begin{cases} \frac{dx}{dt}=4 \\ \frac{dy}{dt}=-8 \end{cases}\) -> direction vector \((4,-8)\) c. \(\begin{cases} \frac{dx}{dt}=3t^2 \\ \frac{dy}{dt}=-3t^2 \end{cases}\) -> direction vector \((3t^2,-3t^2)\) Now that we have the direction vectors and points for each equation, let's compare them.

Compare direction vectors and points

To be describing the same line, both the direction vectors and points must be the same. a. point: \((3,4)\), direction vector: \((1,-2)\) b. point: \((3,4)\), direction vector: \((4,-8)\) c. point: \((3,4)\), direction vector: \((3t^2,-3t^2)\) Notice that the direction vector for b is 4 times the direction vector for a, which means they are parallel and have the same direction. They also share the same point \((3,4)\). Therefore, equations a and b describe the same line. Equation c has a direction vector dependent on t, which means it doesn't have a fixed direction vector. Therefore, equation c does not represent the same line. Finally, the answer is: Equations a and b describe the same line.

Key concepts

Direction Vectors

In parametric equations, direction vectors play a crucial role. They help determine how a line or curve is oriented in space. A direction vector is derived by differentiating each component of the parametric equations with respect to the parameter, typically denoted as \( t \). For instance, if we have the equation \( x = 3 + t \), differentiating it with respect to \( t \) gives \( \frac{dx}{dt} = 1 \). For \( y = 4 - 2t \), the differentiation gives \( \frac{dy}{dt} = -2 \).

Direction vectors indicate how the line travels through space in relation to the parameter \( t \). They are pivotal in understanding the behavior of parametric lines because:

• If two or more equations have proportional direction vectors, they describe lines that are parallel.
• If the points they pass through are identical, then the lines are coincident.

For example, in our given equations, the direction vector for equation a is \((1, -2)\) and for equation b, it is \((4, -8)\). These vectors are proportional \((1:4)\), indicating parallelism.

Lines in Planes

Lines in the Cartesian plane can be represented differently depending on the format of the equation used. Parametric equations are one such format, offering a unique approach by expressing the coordinates \(x\) and \(y\) in terms of a third variable, \(t\). This method allows for easy computation of points along the line.

But how do we know if two parametric equations describe the same line? We need to assess both their direction vectors and the points through which they pass. If their direction vectors are proportional and they share a common point, they describe the same line in the plane.

• Parametric equations such as \( x = 3 + t \), \( y = 4 - 2t \) (equation a) and \( x = 3 + 4t \), \( y = 4 - 8t \) (equation b) illustrate this principle. They share the point \((3,4)\) and have direction vectors \((1, -2)\) and \((4, -8)\), which are scalar multiples of each other.
• This commonality confirms that equations a and b are, in fact, representations of the same line.

Thus, understanding parametric lines requires scrutiny of both the proportionality of direction vectors and the coincidence of points.

Vector Calculus

Vector calculus offers tools that help us navigate problems involving vectors like those posed by parametric equations. It provides operations such as differentiation, which is employed to find direction vectors from parametric equations. By treating each coordinate as a separate function of the parameter \( t \), differentiation helps determine how fast and in what direction the line is moving at any given point.

The connection of vector calculus to parametric equations is significant due to its applications:

• Differentiation of the parametric equations yields the direction vectors necessary to determine line characteristics.
• Vector calculus aids in identifying whether two vectors are proportional and therefore whether they indicate parallelism between lines.

This branch of mathematics therefore acts as an essential background, offering a deeper understanding of geometric places and linearity through parametric equations. Understanding these principles can greatly aid in solving similar exercises, making the learning process smoother and more intuitive.