12.2
Many real-world motions, like a plane’s flight path, involve movement in three-dimensional space. These paths can be modeled using space curves.
A space curve is the set of all points traced by a moving particle as the parameter t varies over an interval. Its position is given by a vector function r(t), where x, y, and z are differentiable functions of t.
A common example is a 3D helix, a smooth curve that spirals upward. It can be described by a vector function with cos(t) in the x-direction, sin(t) in the y-direction, and t in the z-direction.
At t equals zero, the vector simplifies to one, zero, zero, which is the starting point of the curve.
As t increases, the x and y components trace a circle when viewed from above because the cosine of t and the sine of t define circular motion parametrically. At the same time, the z-value increases steadily, causing the curve to rise.
Together, these components form a spiral that wraps around a cylinder.
This shows how a space curve can model a plane’s position as it moves through three-dimensional space.
A space curve describes the path followed by a particle moving through three-dimensional space. Unlike plane curves, which are confined to two coordinates, space curves require three coordinate functions. If t is a parameter, the position of the particle is represented by the vector function
\begin{equation*}\mathbf{r}(t)=\langle x(t),y(t),z(t)\rangle,\end{equation*}
where x(t), y(t), and z(t) are differentiable functions of t. As t varies over an interval, the endpoints of the position vectors trace the curve.
A standard example of a space curve is the circular helix, defined by\begin{equation*}\mathbf{r}(t)=\langle \cos (t),\sin (t),t\rangle.\end{equation*}At t = 0, the position vector is\begin{equation*}\mathbf{r}(0)=\langle 1,0,0\rangle,\end{equation*}So the curve begins at the point (1, 0, 0). As t increases, the x- and y-coordinates are determined by cos(t) and sin(t). These functions satisfy the identity\begin{equation*}x^2+y^2=\cos^2 (t)+\sin^2 (t)=1,\end{equation*}showing that the projection of the curve onto the xy-plane is a circle of radius 1.
While the x- and y-coordinates produce circular motion, the z-coordinate increases linearly as z = t. This steady vertical rise causes the curve to spiral upward rather than remain in a plane. The resulting path wraps around the cylinder, forming a smooth three-dimensional helix.
Space curves are useful for modeling physical motion, such as the flight path of an aircraft, because they describe position in all three spatial directions simultaneously.
Many real-world motions, like a plane’s flight path, involve movement in three-dimensional space. These paths can be modeled using space curves.
A space curve is the set of all points traced by a moving particle as the parameter t varies over an interval. Its position is given by a vector function r(t), where x, y, and z are differentiable functions of t.
A common example is a 3D helix, a smooth curve that spirals upward. It can be described by a vector function with cos(t) in the x-direction, sin(t) in the y-direction, and t in the z-direction.
At t equals zero, the vector simplifies to one, zero, zero, which is the starting point of the curve.
As t increases, the x and y components trace a circle when viewed from above because the cosine of t and the sine of t define circular motion parametrically. At the same time, the z-value increases steadily, causing the curve to rise.
Together, these components form a spiral that wraps around a cylinder.
This shows how a space curve can model a plane’s position as it moves through three-dimensional space.
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