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Chapter 8.3 – Vector, Parametric and Symmetric Equations of a Line in R³
Example: A line passes through [pic]and [pic].
a) Determine a valid direction vector for the line
b) Determine a valid vector equation for the line
c) Determine a valid parametric equation for the line
Solution:
|Generally in R³, a line that intersects the point [pic]with a direction vector [pic]has |
|• a vector equation of [pic], or expressed another way |
|[pic] |
|• a parametric equation of |
|[pic] |
|[pic] |
|[pic] , [pic] |
Example: Show that the following two lines represent the same line
[pic] and
[pic]
Solution:
Example: Show that the following are two different lines
[pic]
Solution:
Symmetric Equation of a Line in R³
Refer to the parametric equations of a line in R³ as indicated in the box in the middle of the previous page.
|[pic] |[pic] |[pic] |
Since each column equals t, we can state that
| | |
|[pic] , [pic] | |
| |← Symmetric equation of a line in R³ |
| |Teacher, show that by definition a,b,c are d1, |
| |d2, d3, ie., the components of the direction |
| |vector |
Example: Determine the symmetric equation of the line in each of the following situations:
a) the line goes through the points [pic]and [pic]
b) the line goes through the points [pic]and [pic]
Solution:
................
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