91Ó°ÊÓ

Parametric equations provide a unique way to describe curves using parameters, commonly represented by a variable like \( t \). In a parametric equation system, each component of a vector function depends on a parameter, dictating how points on a curve are generated. In our context, the vector function \( \mathbf{r}(t) = 2 \mathbf{i} + t \mathbf{j} \) is a simple parametric equation.

This particular equation has two components:

• The \( x \)-component is constant, \( x = 2 \).
• The \( y \)-component changes linearly with \( t \), expressed as \( y = t \).

This combination indicates a straight vertical line on the coordinate plane because the \( x \)-component remains unchanged while the \( y \)-component varies. In essence, parametric equations allow us to express geometric objects more flexibly and intuitively, which can make complex curves much more manageable.

Vector functions are mathematical expressions where each term is itself a vector. They are useful for describing movement and change in multidimensional space by assigning a vector to every value of a parameter, typically \( t \). Our vector function, \( \mathbf{r}(t) = 2 \mathbf{i} + t \mathbf{j} \), assigns vectors in a two-dimensional space.

In the given exercise, the vector function is broken down as follows:

• \( \mathbf{i} \) is the unit vector in the \( x \)-direction. As the coefficient is 2, this indicates all vectors of this function have an unchanging \( x \)-component.
• \( \mathbf{j} \) is the unit vector in the \( y \)-direction. The coefficient \( t \) implies that as \( t \) varies, the \( y \)-component evolves linearly with \( t \).

This straight line alignment is a fundamental trait of vector functions when analyzing linear parametric equations. Vector functions are essential for modeling real-world phenomena like projectile motion and fluid dynamics, providing a clear and concise representation of physical systems.

Coordinate planes are tools for graphically representing mathematical equations in a two-dimensional space, consisting primarily of an \( x \)-axis and a \( y \)-axis. They are fundamental in visualizing how equations translate into geometric shapes.

In this problem, we use a coordinate plane to visually express the path traced by the vector function \( \mathbf{r}(t) = 2 \mathbf{i} + t \mathbf{j} \). Here's how it plots:

• The line is vertical through all points where \( x = 2 \).
• Since the \( y \)-component corresponds to \( y = t \), as \( t \) moves, the line extends indefinitely upward and downward.

Using a coordinate plane, sketching this becomes straightforward: draw a vertical line that crosses the x-axis at point 2. To indicate increasing \( t \), mark arrows pointing upward alongside the line. This visualization helps make abstract vector functions more tangible, allowing easier analysis and comprehension of mathematical relationships and behaviors.