# 10.5 - Eq. of Lines & Planes

- StudyBlue
- North-carolina
- University of North Carolina - Charlotte
- Mathematics
- Mathematics 2241
- Gordon
- 10.5 - Eq. of Lines & Planes

**Created:**2016-12-21

**Last Modified:**2016-12-23

**Vector Equation**of L:

**r**_{0}=**r**+**v****r**=**r**_{0}+*t***v***t*=**r**_{0}×**r**

**r**=

**r**

_{0}+

*t*

**v**

**v**that gives the direction of the line

*L*is written in component form as

**v**=〈

*a,b,c*〉, then we have

*t*

**v**=

- 〈
*ta,tb,tc*〉 - 〈
*a,b,c*〉

*ta,tb,tc*〉

**r**= ________ and

**r**

_{0}= _______

**r**=〈*a,b,c*〉;**r**_{0}=〈*a*〉_{1},b_{2},c_{3}**r**=〈*x,y,z*〉;**r**_{0}=〈*x*〉_{0},y_{0},z_{0}

**r**=〈

*x,y,z*〉;

**r**

_{0}=〈

*x*〉

_{0},y_{0},z_{0}**r**=〈

*x,y,z*〉&

**r**

_{0}=〈

*x*〉, so the vector eq. becomes:

_{0},y_{0},z_{0}- 〈x,y,z〉= 〈x0 − ta, y0 − tb, z0 − tc〉
- 〈x,y,z〉= 〈x0 + ta, y0 + tb, z0 + tc〉

*x,y,z*〉= 〈

*x*〉

_{0}+ ta**,**y_{0}+ tb**,**z_{0}+ tc**r**=

**r**

_{0}+

*t*

**v***x,y,z*〉= 〈

*x*〉

_{0}+ ta**,**y_{0}+ tb**,**z_{0}+ tc- equal
- different

*x,y,z*〉= 〈

*x*〉

_{0}+ ta**,**y_{0}+ tb**,**z_{0}+ tc*x = x*= ????

_{0}+ at, y = y_{0}+ bt, z*z = y*_{0}+ at*z = z*_{0}+ ct

*z = z*

_{0}+ ct*x = x*

_{0}+ at*y = y*

_{0}+ bt*z = z*

_{0}+ ct_{0}+ at

_{0}+ bt

_{0}+ ct

- parametric equations
- symmetric equations

**parametric equations**of the line

*L*through the point

*P*and parallel to the vector

_{0}(x_{0}, y_{0}, z_{0})**v**=〈

*a, b, c*〉.

**v**=〈

*a, b, c*〉is used to describe the direction of a line L, then the numbers

*a, b,*and

*c*are called __________________ of

*L*.

- parametric equations
- directional numbers

*x - x*

_{0})/ a = (y - y_{0})/ b = (z - z_{0})/ c- parametric equations
- symmetric equations

_{0}to r

_{1}is given by the vector equation:

**r**(*t*) = (1 −*t*)**r**_{0}+*t***r**_{1}, where 0 ≤*t*≤ 1**r**(*t*) = (1 +*t*)**r**_{0}−*t***r**_{1}, where 0 >*t*> 1

**r**(

*t*) = (1 −

*t*)

**r**

_{0}+

*t*

**r**

_{1}, where 0 ≤

*t*≤ 1

**lines**are lines that do not intersect and are not parallel (and therefore do not lie in the same plane.)

- Skew
- Perpendicular

**n**is called a ______ vector.

- scalar
- normal

**n**is orthogonal to:

**r**+**r**_{0}**r**-**r**_{0}**r**_{0}-**r**

**r**-

**r**

_{0}

**n**• (

**r**-

**r**

_{0}) = 0

**n**•

**r**=

**n**•

**r**

_{0}

- scalar equation of the plane
- vector equation of the plane

**n**=?,

**r**=?,

**r**

_{0 }=?

**n**=〈a,b,c〉**r**= 〈x,y,z〉**r**_{0}= 〈x_{0},y_{0},z_{0}〉**n**=〈x,y,z〉**r**= 〈a,b,c〉**r**_{0}= 〈x_{0},y_{0},z_{0}〉

**n**=〈a,b,c〉

**r**= 〈x,y,z〉

**r**

_{0}= 〈x

_{0},y

_{0},z

_{0}〉

**n**=〈a,b,c〉

**r**= 〈x,y,z〉

**r**

_{0}= 〈x

_{0},y

_{0},z

_{0}〉becomes:

**n**• (

**r**-

**r**

_{0}) which becomes:

_{0}, y − y

_{0}, x − x

_{0}〉= 0, or:

- a(x − x
_{0}) + b(y − y_{0}) + c(z − z_{0}) = 0 - a(x − x
_{0}) − b(y − y_{0}) − c(z − z_{0}) = 0

_{0}) + b(y − y

_{0}) + c(z − z

_{0}) = 0

_{0}) + b(y − y

_{0}) + c(z − z

_{0}) = 0

- scalar equation of the plane
- vector equation of the plane

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