12.5 Lines and Curves in Space

Question 1

What is a vector valued function?

Answer 1

A function of the form r(t) =(x(t),y(t),z(t))

• A set of three parametric equations that describes a curve in space.
• Meaning three dependent variables (x, y and z) are the components of r, and each part varies with respect to a single independent variable t.

Used to describe the full trajectory of an object as it unfolds with time.

Question 2

Draw a graph describing vector-valued functions.

Answer 2

As t varies, a point (x(t), y(t), z(t)) on a parametric curve is also the head of the position r(t)=(x(t),y(t),z(t)).

Question 3

What determines a unique line.

Answer 3

Two distinct points in R³ determine a unique line. Also, one point and a direction determine a unique line.