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Chapter 12: Problem 7

Find the parametric equations that correspond to the given vector equation. $$ \mathbf{r}=3 t^{2} \mathbf{i}-2 \mathbf{j} $$

Short Answer

Expert verified
The parametric equations are \( x(t) = 3t^2 \) and \( y(t) = -2 \).

Step by step solution

01

Identify the Components

The given vector equation is \( \mathbf{r} = 3t^2 \mathbf{i} - 2 \mathbf{j} \). This vector is comprised of two components: one along the \( \mathbf{i} \) direction (the x-component) and one along the \( \mathbf{j} \) direction (the y-component).
02

Extract the X-Component

The x-component of the vector equation is the term associated with \( \mathbf{i} \). From the given equation \( \mathbf{r} = 3t^2 \mathbf{i} - 2 \mathbf{j} \), the x-component is \( x = 3t^2 \).
03

Extract the Y-Component

The y-component of the vector equation is the term associated with \( \mathbf{j} \). From the equation \( \mathbf{r} = 3t^2 \mathbf{i} - 2 \mathbf{j} \), the y-component is \( y = -2 \).
04

Write the Parametric Equations

Using the components extracted in the previous steps, we can write the parametric equations as: \( x = 3t^2 \) for the x-direction and \( y = -2 \) for the y-direction. Therefore, the parametric equations are: \[ x(t) = 3t^2, \, y(t) = -2 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector Equations
In mathematics, a vector equation describes a line or a curve in space by tracing points using a vector expression. A vector equation typically represents the position of a point as a function of a parameter, often denoted by \( t \). It consists of different components, each affecting a directional vector. In our example, the vector equation is given as \( \mathbf{r} = 3t^2 \mathbf{i} - 2 \mathbf{j} \).
  • The term \( 3t^2 \mathbf{i} \) pertains to movement in the x-direction.
  • The term \( -2 \mathbf{j} \) indicates movement in the y-direction.
These components tell us how the point moves in each direction with each change in the parameter \( t \). Understanding vector equations is crucial for translating vector notations into more familiar parametric equations.
X-Component
The x-component of a vector equation tells us how far along the x-axis the points on the line or curve are for any particular value of the parameter. For the vector equation \( \mathbf{r} = 3t^2 \mathbf{i} - 2 \mathbf{j} \), the x-component is extracted by isolating the part associated with \( \mathbf{i} \).
Here, the term is \( 3t^2 \mathbf{i} \), leading to the x-component as \( x = 3t^2 \). This expression means that the x-coordinate of a point on our line changes as the square of the parameter \( t \). It shows us that as the value of \( t \) varies, the line moves horizontally in a parabolic manner dictated by the function \( 3t^2 \).
Y-Component
Just as the x-component provides horizontal information, the y-component offers a vertical description of our point's movement. Looking again at our vector equation \( \mathbf{r} = 3t^2 \mathbf{i} - 2 \mathbf{j} \), the component linked to \( \mathbf{j} \) is \( -2 \mathbf{j} \). From this, we can see that the y-component is simply \( y = -2 \).
Unlike the x-component, the value of \( y \) is constant here, meaning no matter what \( t \) is, the y-coordinate remains fixed at \( -2 \). This effectively makes our line or curve reside along a constant vertical level or y-value, translating into a horizontal line in parameter space.
Parametrization
Parametrization is the process of selecting a parameter to describe a curve or surface mathematically. It allows us to write equations that map a range of values into coordinates. In this example, the parametric equations that arise from our vector equation are essentially splitting the vector into two separate equations based on its components.
  • The x-direction equation becomes \( x(t) = 3t^2 \).
  • The y-direction equation is \( y(t) = -2 \).
These expressions detail the trajectory of a point by assigning each coordinate a unique function of \( t \); here, \( t \) is the parameter. Parametrization is highly valuable because it can describe paths, enable calculating line integrals, and aid in visualizing planes and surfaces across physics and mathematics.

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Most popular questions from this chapter

Determine whether the statement is true or false. Explain your answer. If \(\mathbf{r}(s)\) parametrizes the graph of \(y=|x|\) in 2 -space by arc length, then \(\mathbf{r}(s)\) is smooth.

These exercises are concerned with the problem of creating a single smooth curve by piecing together two separate smooth curves. If two smooth curves \(C_{1}\) and \(C_{2}\) are joined at a point \(P\) to form a curve \(C\), then we will say that \(C_{1}\) and \(C_{2}\) make a smooth transition at \(P\) if the curvature of \(C\) is continuous at \(P .\) Show that the transition at \(x=0\) from the horizontal line \(y=0\) for \(x \leq 0\) to the parabola \(y=x^{2}\) for \(x>0\) is not smooth, whereas the transition to \(y=x^{3}\) for \(x>0\) is smooth.

Find \(\mathrm{T}(t)\) and \(\mathrm{N}(t)\) at the given point. $$ x=\cosh t, y=\sinh t, z=t ; t=\ln 2 $$

Find an arc length parametrization of the curve that has the same orientation as the given curve and for which the reference point corresponds to \(t=0 .\) $$ \mathbf{r}(t)=\frac{1}{3} t^{3} \mathbf{i}+\frac{1}{2} t^{2} \mathbf{j} ; t \geq 0 $$

A shell is to be fired from ground level at an elevation angle of \(30^{\circ}\). What should the muzzle speed be in order for the maximum height of the shell to be \(2500 \mathrm{ft}\) ?
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