13.1 - Vector Functions

What is a Vectors Function?

The following equation represents the form of a standard vector function

$r (t) = ⟨ x (t), y (t), z (t)⟩$

Where inputs are real numbers, and outputs are vectors.

(similar to parametric equations from calculus 2)

Sometimes, t is restricted to some parameter domain, eg:

$0 \leq t \leq 2 π$

$t \in [0, 2 π]$

Lets graph a vector function Where x = cos t a y = sin t between 0 and infinity

This equation yields a “helix” shape, one of the primary shapes that come from a vector function

Example 2: Graph this equation:

$r (t) = ⟨ x_{0}, y_{0}, z_{0} ⟩ + t ⟨ a, b, c ⟩$

$l in e = ⟨ x_{0} + a t, y_{0} + b t, z_{0} + c t ⟩$

For the line segment p to q, using example NumPy arrays:

p = np.array([1, 2, 3])
q = np.array([4, 5, 6])

Resultant Graph

Find a vector function (parameterization) of the curve of intersection of the surfaces:

$x^{2} + y^{2} = 1$

$y + z = 2$

Visualization of the following equations

$4 y = x^{2} + z^{2}$

$y = x$

2d Rendering: