Now, evaluate \( \vec{F}(\vec{r}(t)) = \langle -3\sin t, 3\cos t \rangle \)

["Evaluating the Vector Function ( \vec{F}(\vec{r}(t)) = \langle -3\sin t, 3\cos t \rangle ): A Comprehensive Overview", "When analyzing the vector function ( \vec{F}(\vec{r}(t)) = \langle -3\sin t, 3\cos t \rangle ), it’s essential to understand both its geometric interpretation and its physical meaning—especially in contexts such as motion, physics, and vector calculus. This article explores how to evaluate and interpret this vector field, its properties, and its applications.", "---", "### What Is ( \vec{F}(\vec{r}(t)) = \langle -3\sin t, 3\cos t \rangle )?", "The given function defines a vector-valued function of a single real parameter ( t ), commonly interpreted as time. It maps each real value ( t \in \mathbb{R} ) to a two-dimensional vector in ( \mathbb{R}^2 ):", "[\n\vec{F}(t) = \langle -3\sin t,\ 3\cos t \rangle\n]", "Geometrically, this vector function traces out a curve in the plane as ( t ) varies over the real numbers.", "---", "### Evaluating the Vector Function", "To evaluate ( \vec{F}(t) ), simply substitute a specific value of ( t ). For example, let’s compute the vector at ( t = 0 ):", "[\n\vec{F}(0) = \langle -3\sin(0),\ 3\cos(0) \rangle = \langle 0,\ 3 \rangle\n]", "Similarly, at ( t = \frac{\pi}{2} ):", "[\n\vec{F}\left(\frac{\pi}{2}\right) = \langle -3\sin\left(\frac{\pi}{2}\right),\ 3\cos\left(\frac{\pi}{2}\right) \rangle = \langle -3,\ 0 \rangle\n]", "At ( t = \pi ):", "[\n\vec{F}(\pi) = \langle -3\sin(\pi),\ 3\cos(\pi) \rangle = \langle 0,\ -3 \rangle\n]", "At ( t = \frac{3\pi}{2} ):", "[\n\vec{F}\left(\frac{3\pi}{2}\right) = \langle -3\sin\left(\frac{3\pi}{2}\right),\ 3\cos\left(\frac{3\pi}{2}\right) \rangle = \langle 3,\ 0 \rangle\n]", "These evaluations generate key points on the trajectory: ( (0,3) ), ( (-3,0) ), ( (0,-3) ), and ( (3,0) ), forming a path in the plane.", "---", "### Geometric Interpretation: A Circular Trajectory?", "By recalling trigonometric identities, observe:", "[\nx(t) = -3\sin t,\quad y(t) = 3\cos t\n]", "We can eliminate ( t ) to find the Cartesian equation:", "[\n\sin t = -\frac{x}{3},\quad \cos t = \frac{y}{3}\n]", "Using ( \sin^2 t + \cos^2 t = 1 ):", "[\n\left(-\frac{x}{3}\right)^2 + \left(\frac{y}{3}\right)^2 = 1 \quad \Rightarrow \quad \frac{x^2}{9} + \frac{y^2}{9} = 1\n]", "Multiplying by 9:", "[\nx^2 + y^2 = 9\n]", "This is the equation of a circle centered at the origin with radius 3.", "Since the parametrization uses ( t ), and the trigonometric functions are periodic with period ( 2\pi ), the curve ( \vec{F}(t) ) traces this circle in a counter-clockwise direction as ( t ) increases. However, because ( y(t) = 3\cos t ) decreases as ( t ) increases past 0 (starting at max ( y )), this corresponds to a clockwise traversal of the circle.", "Thus, ( \vec{F}(t) ) traces the circle ( x^2 + y^2 = 9 ) clockwise once per period ( 2\pi ).", "---", "### Physical and Vector Calculus Interpretation", "From a physics perspective, vector fields like ( \vec{F}(t) ) often represent forces or velocity fields depending on context. Here, interpreted as a velocity vector, this motion lies on a circular path with constant speed.", "Compute the speed by finding the magnitude of the vector:", "[\n|\vec{F}(t)| = \sqrt{(-3\sin t)^2 + (3\cos t)^2} = \sqrt{9\sin^2 t + 9\cos^2 t} = \sqrt{9(\sin^2 t + \cos^2 t)} = \sqrt{9} = 3\n]", "Thus, the magnitude is constant: speed = 3 units per time.", "To find the direction of motion, note the derivative of positionor velocity vector (if ( \vec{F} = \vec{v}(t) ), the velocity):", "[\n\vec{v}(t) = \frac{d}{dt} \langle -3\sin t, 3\cos t \rangle = \langle -3\cos t,\ -3\sin t \rangle\n]", "The direction of motion at any ( t ) is given by ( \vec{v}(t) ), indicating tangential velocity consistent with circular motion.", "---", "### Parametric to Cartesian Conversion and Applications", "The parametric form elegantly connects to trigonometric scaling and rotation. Such parameterizations are common in:", "- Modeling periodic motion, such as orbital paths\n- Designing circular motion controllers in robotics\n- Visualizing harmonic oscillators in physics", "Moreover, rewriting ( \vec{F}(t) ) reveals rotational symmetry, useful in vector calculus for computing line integrals, curls (showing zero curl = irrotational field), or flux through circular regions.", "---", "### Summary", "Evaluating ( \vec{F}(\vec{r}(t)) = \langle -3\sin t, 3\cos t \rangle ) reveals a unit-speed circling motion along a circle of radius 3 centered at the origin, traced clockwise as ( t ) increases. The algebra confirms this via the circle equation, while vector analysis confirms constant speed and tangential velocity. Understanding such functions is pivotal in physics, engineering, and geometric modeling.", "---", "### Key Takeaways", "- Evaluation: Plug in values of ( t ) to get discrete points on a circular path.\n- Geometric shape: Circle of radius 3 centered at origin.\n- Direction: Traversed clockwise with magnitude 3.\n- Applications: Ornithology, robotics, electromagnetism, and oscillatory systems.", "---", "For further exploration, consider converting this to polar coordinates or analyzing its derivative to compute curvature and arc length—valuable tools in advanced vector analysis."]