Equations of Lines and Planes in Math 114 – Rimmer - Prof. E. So, Study notes of Mathematics

Download Equations of Lines and Planes in Math 114 – Rimmer - Prof. E. So and more Study notes Mathematics in PDF only on Docsity!

13.5 Equations of Lines and Planes 13.5 Equations of Lines and Planes In order to find the equation of a line, we need :

A ) a point on the line P 0 ( x 0 , y 0 , z 0 )

B ) a direction vector for the line v = a b c , ,

x

y

z

P 0 (^) ( x 0 (^) , y 0 (^) , z 0 )

0 v

r

r

P x y z ( , ,)

P P 0 = t v




r = r 0 (^) + t v

r = x y z , ,

r 0 = x 0 (^) , y 0 (^) , z 0

P P 0 = t v = t a b c , ,



L

vector equation of line L

x y z , , = x 0 (^) , y 0 (^) , z 0 + t a b c , ,

x = x 0 (^) + at , y = y 0 (^) + bt , z = z 0 + ct parametric equations of the line L

equating components:

eliminating the parameter t (solve for t in each, then equate the results) x x 0 (^) y y 0 (^) z z 0 a b c

− − − = = symmetric equations of the line L

Math 114 – Rimmer 13.5 Equations of Lines and Planes

Find parametric equations of the line containing ( 5,1,3) and ( 3, −2, 4 .)

Two lines in 3 space can interact in 3 ways: 13.5 Equations of Lines and Planes

A) Parallel Lines - their direction vectors are scalar multiples of each other

B) Intersecting Lines - there is a specific t and s , so that the lines share the same point.

C) Skew Lines - their direction vectors are not parallel and there is no values of t and s that make the lines share the same point.

Math 114 – Rimmer Determine whether the lines 1 and 2 are parallel, skew 13.5 Equations of Lines and Planes or intersecting. If they intersect, find the point of intersection.

L L

Set the x s ' =

Set the y s ' =

3 − t = 8 + 2 s − = t 5 + 2 s t = − 5 − 2 s

5 + 3 t = − 6 − 4 s 3 t = − 11 − 4 s 11 4 3

s t

− −

=

This should happen at the same time , so t set these equal. 11 4 5 2 3

s s

− −

− − =

− 15 − 6 s = − 11 − 4 s − 6 s = 4 − 4 s − 2 s = 4 s = − 2

t = − 5 − 2 ( − 2 )

t = − 1

Check to make sure that the values are equal for this and.

z t s − − 1 4 t = 5 + s

− − 1 4 ( − 1 ) = 5 + ( − 2 )

3 =3 check Now find the pt. of intersection.

( ) ( ) ( )

using 1 3 1 5 3 1 1 4 1

L x y z

= − − = + − = − − −

( 4, 2,3)

1 3 5 3 1 4

L x t y t z t

= − = + = − −

2 8 2 6 4 5

L x s y s z s

= + = − − = +

Determine whether the lines 1 and 2 are parallel, skew 13.5 Equations of Lines and Planes or intersecting. If they intersect, find the point of intersection.

L L

1 4 8 2 12

L x t y t z t

= + = − − =

2 3 2 1 3 3

L x s y s z s

= + = − + = − −

Math 114 – Rimmer 13.5 Equations of Lines and Planes In order to find the equation of a plane, we need :

A ) a point on the plane P 0 ( x 0 , y 0 , z 0 )

B ) a vector that is orthogonal to the plane n = a b c , ,

Planes

this vector is called the to the plane

normal vector

x

y

z

P 0 (^) ( x 0 (^) , y 0 (^) , z 0 )

r 0

r

P x y z ( , ,)

P P 0 = r − r 0



n

n ⋅ (^) ( r − r 0 )= 0 n r ⋅ = n r ⋅ 0 vector equation of the plane

r = x y z , ,

r 0 = x 0 (^) , y 0 (^) , z 0

P P 0 = x − x 0 (^) , y − y 0 (^) , z − z 0



n = a b c , ,

( )

( ) ( ) ( )

0 0 0 0

0 a x x b y y c z z 0

⋅ − = ⇒ − + − + − =

n r r

scalar equation of the plane

ax − ax 0 (^) + by − by 0 (^) + cz − cz 0 = 0

ax + by + cz − ( ax 0 + by 0 + cz 0 )= 0

ax + by + cz + d = 0 linear equation of the plane

Determine the equation of the plane that contains the lines L 1 (^) and L 2. 13.5 Equations of Lines and Planes

1 3 5 3 1 4

L x t y t z t

= − = + = − −

2 8 2 6 4 5

L x s y s z s

= + = − − = +

In order to find the equation of a plane, we need :

A ) a point on the plane

B ) a vector that is orthogonal to the plane

1 (^ )^2 (^ ) n = a b c^ , ,

We have two points in the plane: from L : 3,5, − 1 and from L : 8, −6,

1 2

We have two vectors in the plane: from L : −1,3, − 4 and from L : 2, −4,

Math 114 – Rimmer 13.5 Equations of Lines and Planes ( ) ( ) ( )

Determine the equation of the plane that passes through 1, 2,3 , 3, 2,1 , and −1, −2, 2.

Two distinct planes in 3-space either are 13.5 Equations of Lines and Planes parallel or intersect in a line.

x

y

z

Math 114 – Rimmer Find the line of intersection of the two planes 13.5 Equations of Lines and Planes 2 0 2 3 2 0

x y z x y z

− + =

• − =

13.5 Equations of Lines and Planes

If two planes intersect, then you can determine the angle between them.

 between planes = between their normal vectors

n 2 n^1

1 2 1 2

cos θ

⋅

=

n n

n n

θ

θ

Find the between the planes 2 0 2 3 2 0

x y z x y z

− + =

• − =

Math 114 – Rimmer

Distance between a point and a plane: 13.5 Equations of Lines and Planes

D comp

⋅

= n =

n b

b

n

1 1 1 2 2 2

ax by cz d

a b c

+ + +

=

+ +

b

n

D