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

Jason S.

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

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If the vector **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*〉

We can also write **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}

We can write **r** =〈*x,y,z*〉& **r**_{0} =〈*x*_{0},y_{0},z_{0}〉, so the vector eq. becomes:

- 〈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〉

Two vectors are equal if and only if corresponding components are ________.

- equal
- different

equal

〈*x,y,z*〉= 〈*x*_{0} + ta **,** y_{0} + tb **,** z_{0} + tc〉

We have 3 scalar eq.:*z = y*_{0}+ at*z = z*_{0}+ ct

x = x_{0} + at

y = y_{0} + bt

z = z_{0} + ct

These are called

- parametric equations
- symmetric equations

parametric equations

These equations are called **parametric equations** of the line *L* through the point *P*_{0}(x_{0}, y_{0}, z_{0}) and parallel to the vector **v** =〈*a, b, c*〉.

In general, if a vector **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

direction numbers

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

These are called:

- parametric equations
- symmetric equations

symmetric equations

The line segment from r_{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

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

- Skew
- Perpendicular

Skew

The orthogonal vector **n** is called a ______ vector.

- scalar
- normal

normal

In particular, **n** is orthogonal to:

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

Are both called:

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

vector equation of the plane

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To obtain a scalar equation for the plane, we write **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}〉

〈a,b,c〉•〈x − x_{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

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

Is called:

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

scalar equation of the plane

... with normal vector n = 〈a,b,c〉

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