3-Dimensional lines
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Intros
Lessons
- 3-Dimensional Lines Overview:
- 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>
- Parametric Equations
- Another way to express 3D lines
- How to get Parametric Equations
- An example
- Symmetric Equations
- Another way to express 3D lines
- How to get Symmetric Equations?
- An example
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Examples
Lessons
- Question with Two Points
If the line passes through the points (2,1,0) and (1,4,5), find the equation of the line in vector form. - If the line passes through the points (−2,3,5) and (−1,0,−3), find the equation of the line in symmetric form.
- Question with a Point and a Line
If the line passes through the point (0,1,3) and is parallel to the line <1−2t,5+t,−t>, find the vector equation of the line. - Intersection of a Line and a Plane
Determine whether the line given by <5−t,−5,1+2t> and the xy-plane will intersect. If so, find the intersection point. - Determine whether the line given by <5−t,−5,1+2t> and the xz-plane will intersect. If so, find the intersection point.
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Topic Notes
Notes:
Vector Equation for 3-Dimensional Lines
Unlike 2D lines which have the equation y=mx+b, 3-Dimensional lines can be expressed as vector functions in the form
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:
Symmetric Equations Assume that a,b,c are non-zero. Then rearranging the set of equations from above gives us:
Vector Equation for 3-Dimensional Lines
Unlike 2D lines which have the equation y=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>
where t is a variable. Here is the visual representation of the vector r(t).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>
<x,y,z>=<x0,y0,z0>+<ta,tb,tc>
=<x0+ta,y0+tb,z0+tc>
x=x0+ta
y=y0+tb
z=z0+tc
Symmetric Equations Assume that a,b,c are non-zero. Then rearranging the set of equations from above gives us:
x=x0+ta→x−x0=ta→ax−x0=t
y=y0+tb→y−y0=tb→→by−y0=t
z=z0+tc→z−z0=tc→cz−z0=t
ax−x0=by−y0=cz−z0
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