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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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