3-Dimensional lines

3-Dimensional lines

Lessons

Notes:

Vector Equation for 3-Dimensional Lines
Unlike 2D lines which have the equation y=mx+by=mx+b, 3-Dimensional lines can be expressed as vector functions in the form

r(t)=r0+tv=<x0,y0,z0>+t<a,b,c>\vec{r(t)} = \vec{r_0} + \vec{tv} = \lt x_0, y_0, z_0 \gt + \; t \lt a,b,c \gt

where t is a variable. Here is the visual representation of the vector r(t)\vec{r(t)} .
expressing 3d line
There are 2 other alternate ways to express a 3-Dimensional line.

Parametric Equation
From the equation above, we can rearrange the vector equation so that:

r(t)=<x0,y0,z0>+t<a,b,c>\vec{r(t)} = \lt x_0,y_0,z_0 \gt + \;t\lt a,b,c \gt
<x,y,z>=<x0,y0,z0>+<ta,tb,tc>\lt x,y,z \gt = \lt x_0, y_0, z_0 \gt + \lt ta,tb,tc \gt
=<x0+ta,y0+tb,z0+tc>= \lt x_0 + ta, y_0 + tb , z_0 + tc \gt

Then we can see that:

x=x0+tax = x_0 + ta
y=y0+tby = y_0 + tb
z=z0+tcz = z_0 + tc

Hence these set of equations of the Parametric Equation of the equation of the 3D line.

Symmetric Equations Assume that a,b,ca,b,c are non-zero. Then rearranging the set of equations from above gives us:

x=x0+taxx0=taxx0a=t x = x_0 + ta \to x - x_0 = ta \to \frac{x-x_0}{a} = t
y=y0+tbyy0=tbyy0b=ty = y_0 + tb \to y - y_0 = tb \to \to \frac{y-y_0}{b} = t
z=z0+tczz0=tczz0c=tz = z_0 + tc \to z - z_0 = tc \to \frac{z-z_0}{c} = t

Hence we have our symmetric equation of the equation of the 3D line:

xx0a=yy0b=zz0c\frac{x-x_0}{a} = \frac{y-y_0}{b} = \frac{z-z_0}{c}

  • Introduction
    3-Dimensional Lines Overview:
    a)
    Vector Equation 3D lines
    • Need a vector equation to express 3D lines
    • Diagram for the vector equation
    • r(t)=<x0,y0,z0>+t<a,b,c>\vec{r(t)} = \lt x_0 , y_0, z_0 \gt +\; t \lt a,b,c \gt

    b)
    Parametric Equations
    • Another way to express 3D lines
    • How to get Parametric Equations
    • An example

    c)
    Symmetric Equations
    • Another way to express 3D lines
    • How to get Symmetric Equations?
    • An example


  • 1.
    Question with Two Points
    If the line passes through the points (2,1,0)(2, 1, 0) and (1,4,5)(1, 4, 5), find the equation of the line in vector form.

  • 2.
    If the line passes through the points (2,3,5)(-2, 3, 5) and (1,0,3)(-1, 0, -3), find the equation of the line in symmetric form.

  • 3.
    Question with a Point and a Line
    If the line passes through the point (0,1,3)(0, 1, 3) and is parallel to the line <12t,5+t,t><1-2t, 5+t, -t> , find the vector equation of the line.

  • 4.
    Intersection of a Line and a Plane
    Determine whether the line given by <5t,5,1+2t><5-t, -5, 1+2t> and the xyxy-plane will intersect. If so, find the intersection point.

  • 5.
    Determine whether the line given by <5t,5,1+2t><5-t, -5, 1+2t> and the xzxz-plane will intersect. If so, find the intersection point.