Vector Functions in 3D Space with Applications to Motion

Vector Functions

A vector valued function that assigns a vector to each real number in its domain is called a vector function. In three-dimensional space, a vector function can be expressed as

r(t) = \langle x(t),y(t),z(t) \rangle

where $x(t)$ , $y(t)$ , and $z(t)$ are real-valued functions of the parameter (t). The parameter (t) often represents time, making vector functions particularly useful for describing the motion of objects in space.

Derivatives of Vector Functions

The derivative of a vector function $r(t)$ is defined as the vector function. It is obtained by differentiating each component function with respect to $t$ :

r'(t) = \langle x'(t), y'(t), z'(t) \rangle

Integrals of Vector Functions

The integral of a vector function $r(t)$ is defined as the vector function obtained by integrating each component function with respect to $t$ :

\int r(t) \, dt = \langle \int x(t) \,dt, \int y(t) \, dt, \int z(t) \, dt \rangle + C

where (C) is a constant vector of integration.

We can extend the concept of definite integrals and the fundamental theorem of calculus to vector functions as well. If $r(t)$ is continuous on the interval ([a, b]), then

\int_a^b r(t) \, dt = \langle \int_a^b x(t) \, dt, \int_a^b y(t) \, dt, \int_a^b z(t) \, dt \rangle

Normal, Tangent, and Binormal Vectors

We often analyze the properties of the vector functions using special vectors associated with it: the tangent vector, the normal vector, and the binormal vector.

Tangent Vector: The tangent vector $T(t)$ is a unit vector that points in the direction of the curve at a given point. It is defined as

T(t) = \frac{r'(t)}{\|r'(t)\|}

$r'(t)$ is the derivative of the vector function, and $\|r'(t)\|$ is its magnitude. $r'(t)$ will give us the direction of tangential motion, while normalizing it gives us a unit vector.

Normal Vector: The normal vector $N(t)$ is a unit vector that points towards the center of curvature of the curve. It is defined as

N(t) = \frac{T'(t)}{\|T'(t)\|}

Binormal Vector: The binormal vector $B(t)$ is a unit vector that is orthogonal to both the tangent and normal vectors. It is defined as

B(t) = T(t) \times N(t)

These three vectors $T(t), N(t), and B(t)$ form an orthonormal basis known as the Frenet-Serret frame (sometimes called TNB frame), which is useful in analyzing the geometry of curves in three-dimensional space. More to come in Differential Geometry!

Arc Length

We can calculate the arc length of a curve defined by a vector function $r(t)$ from $t=a$ to $t=b$ using the formula:

L = \int_a^b \|r'(t)\| \, dt

where $\|r'(t)\|$ is the magnitude of the derivative of the vector function. This formula gives us the total distance traveled along the curve between the two parameter values.

We can similarly define the arc length function $s(t)$ from a starting point $t=a$ to a variable endpoint $t$ as:

s(t) = \int_a^t \|r'(u)\| \, du

and get the rate of change of arc length with respect to the parameter (t) as:

\frac{ds}{dt} = \|r'(t)\|

This relationship shows that the rate of change of arc length with respect to the parameter (t) is equal to the speed of the object moving along the curve defined by the vector function $r(t)$ .

Curvature and Torsion

The curvature of a curve defined by a vector function $r(t)$ measures how quickly the curve changes direction at a given point. We can calculate the curvature $\kappa(t)$ in many ways.

By defitinion, curvature is given by

\kappa(t) = \left\| \frac{dT}{ds} \right\|

where (T) is the unit tangent vector and (s) is the arc length parameter.

Using the chain rule, we can express curvature in terms of the parameter $t$ as follows:

\kappa(t) = \frac{\|T'(t)\|}{\|r'(t)\|}

We can also compute curvature using the cross product of the first and second derivatives of the vector function and dividing by the cube of the magnitude of the first derivative:

\kappa(t) = \frac{\|r'(t) \times r''(t)\|}{\|r'(t)\|^3}

Torsion $\tau(t)$ measures how much a curve twists out of the plane of curvature. It is defined as

\tau(t) = - \frac{dB}{ds} \cdot N

where (B) is the binormal vector and (N) is the normal vector.

Using the chain rule, we can express torsion in terms of the parameter $t$ as follows:

\tau(t) = - \frac{B'(t) \cdot N(t)}{|r^(t)|}

We can also compute torsion using the scalar triple product of the first, second, and third derivatives of the vector function:

\tau(t) = \frac{(r'(t) \times r''(t)) \cdot r'''(t)}{\|r'(t) \times r''(t)\|^2}

Applications to Motion

Vector functions are particularly useful in describing the motion of objects in three-dimensional space. The position of an object at time $t$ can be represented by a vector function $r(t)$ . The velocity and acceleration of the object can be obtained by differentiating the position vector function.

We define the Tangential and Normal Components of Acceleration as follows:

a(t) = r''(t) = a_T T(t) + a_N N(t)

where (a_T) is the tangential component of acceleration and (a_N) is the normal component of acceleration. They can be computed as:

a_T = \frac{d}{dt} \|r'(t)\| = \\frac{r'(t) \cdot r''(t)}{\|r'(t)\|}

a_N = \kappa(t) \|r'(t)\|^2

These components help us understand how the object is accelerating along its path (tangential acceleration) and how it is changing direction (normal acceleration).