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Vector functions are mathematical expressions that describe vector-valued quantities depending on one or more parameters. They are often used in space and plane representations. A vector function can be written in terms of its individual components. For example, the vector function \( \mathbf{r}(t) = x(t) \mathbf{i} + y(t) \mathbf{j} \) specifies a path in the plane defined by the changes in \( x(t) \) and \( y(t) \).
The given vector \( \mathbf{r} = 2 \sin 3t \mathbf{i} - 2 \cos 3t \mathbf{j} \) consists of two functions: \( x(t) = 2 \sin 3t \) and \( y(t) = -2 \cos 3t \). This representation indicates both the direction in which the point moves over time as \( t \) changes and its position at any given \( t \).
Understanding vector functions helps us visualize how quantities linked with a given parameter evolve, an essential skill in physics and engineering.

The equation of a circle is a fundamental concept in geometry. It is given by the formula \( x^2 + y^2 = r^2 \), where \( r \) is the radius of the circle, translating the geometric property into an algebraic one. Circles are centered at the origin (0,0) unless stated otherwise.
In the provided solution, the parametric equations \( x(t) = 2 \sin 3t \) and \( y(t) = -2 \cos 3t \) describe a circle when plugged into this standard form. The expression \( (2 \sin 3t)^2 + (-2 \cos 3t)^2 \) simplifies to \( 4(\sin^2 3t + \cos^2 3t) \). Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we find that the equation becomes \( 4 = 4 \), indicating a circle with \( r = 2 \).
This parametric circle is centered at the origin, and traversed as the parameter \( t \) varies, demonstrating how the circle is formed dynamically by the vector function.

Trigonometric identities are crucial mathematical tools that simplify complex trigonometric expressions. One of the most pivotal identities in mathematics is \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity helps in transforming parametric forms like those in vector functions into recognizable geometric shapes.
In our exercise, to transition from the parametric to the standard circle equation form, this identity was utilized. By applying it, the expression \( 4(\sin^2 3t + \cos^2 3t) \) reduces to 4, echoing the equality \( x^2 + y^2 = 4 \) or \( r^2 = 4 \).
This straightforward yet profound identity shows a seamless bridge from trigonometry to geometry, unraveling the ways in which parametric equations describe simple geometrical forms like circles. Understanding and applying these identities allow for the conversion of paths or loci described through vector functions into familiar and simplified geometric terms.