PLANES MATH 200 MAIN QUESTIONS FOR TODAY How do we describe planes - - PowerPoint PPT Presentation

PLANES

MATH 200 WEEK 2 - FRIDAY

MATH 200

MAIN QUESTIONS FOR TODAY

▸ How do we describe planes in space? ▸ Can we find the equation of a plane that satisfies certain

conditions?

▸ How do we find parametric equations for the line of

intersection of two (non-parallel) planes?

▸ How do we find the (acute) angle of intersection between

two planes.

MATH 200

DEFINING A PLANE

▸ A line is uniquely defined by two points ▸ A plane is uniquely defined by three non-collinear points ▸ Why “non-collinear”? ▸ Suppose we have three points, A, B, and C in space…

A B C

MATH 200

▸ We can draw vectors between these points. ▸ We can find the vector orthogonal to both of these vectors

using the cross product.

▸ This is called a normal vector (n = <a,b,c>) ▸ How does this help us describe the set of points that

make up the plane?

A B C n

MATH 200

▸ Consider a random point P(x,y,z) on the plane. ▸ If A has components (x1,y1,z1), we can connect A to P with

the vector AP = <x - x1, y - y1, z - z1>

▸ The vectors AP and n must be orthogonal! ▸ n • AP = 0 ▸ <a, b, c> • <x - x1, y - y1, z - z1> = 0

P

A B C n

MATH 200

FORMULA

▸ So, given a point (x1, y1, z1) on a plane and a vector normal

to the plane <a, b, c>, every point (x, y, z) on the plane must satisfy the equation

x x1, y y1, z z1 · a, b, c = 0

▸ We can also multiply this out:

a(x − x1) + b(y − y1) + c(z − z1) = 0

THIS IS CALLED POINT-NORMAL FORM FOR A PLANE

MATH 200

FIND AN EQUATION FOR A PLANE

▸ Let’s find an equation for the plane that contains the

following three points:

▸ A(1, 2, 1); B(3, -1, 2); C(-1, 0, 4) ▸ We need the normal: n ▸ n = AB x AC ▸ AB = <2, -3, 1> and AC = <-2, -2, 3> ▸ n = AB x AC = <-7, -8, -10> ▸ Point-normal using A: -7(x-1) - 8(y-2) - 10(z-1) = 0

A B C n

MATH 200

COMPARING ANSWERS

▸ Notice that we had three easy choices for an equation for the

plane in the last example:

▸ Point-normal using A: -7(x-1) - 8(y-2) - 10(z-1) = 0 ▸ Point-normal using B: -7(x-3) - 8(y+1) - 10(z-2) = 0 ▸ Point-normal using C: -7(x+1) - 8y - 10(z-4) = 0 ▸ We also could have scaled the normal vector we used: ▸ -14(x-1) - 16(y-2) - 20(z-1) = 0 ▸ This makes comparing our answers trickier ▸ To make comparing answers easier, we can put the equations

into standard form

MATH 200

▸ Standard form: ▸ ax + by + cz = d ▸ Multiply out point-normal form and combine constant

terms on the left-hand side

▸ E.g. ▸ Point-normal using A: -7(x-1) - 8(y-2) - 10(z-1) = 0 ▸ -7x + 7 - 8y + 16 - 10z + 10 = 0 ▸ -7x - 8y - 10z = -33 or 7x + 8y + 10z = 33 ▸ Point-normal using C: -7(x+1) - 8y - 10(z-4) = 0 ▸ -7x - 7 - 8y - 10z + 40 = 0 ▸ -7x - 8y - 10z = -33 or 7x + 8y + 10z = 33

MATH 200

EXAMPLE 1

▸ Find an equation for the plane in standard form which

contains the point A(-3,2,5) and is perpendicular to the line L(t)=<1,4,2>+t<3,-1,2>

▸ Since the plane is perpendicular to the line L, its

direction vector is normal to the plane.

▸ n = <3,-1,2> ▸ Plane: 3(x+3) - (y-2) + 2(z-5) = 0 ▸ 3x + 9 - y + 2 + 2z - 10 = 0 ▸ 3x - y + 2z = -1

MATH 200

EXAMPLE 2

▸ Find an equation for the

plane containing the point A(1,4,2) and the line L(t)=<3,1,4>+t<1,-1,2>

▸ We need two things: a

point and a normal vector

▸ Point: we can use A ▸ Normal vector: ??? ▸ Draw a diagram to help

A

L

(3,1,4) <1,-1,2>

n = <?,?,?>

WE’LL PUT ALL THE INFORMATION WE HAVE DOWN WITHOUT ANY ATTENTION TO SCALE OR PROPORTION

MATH 200

▸ To get a normal we could

take the cross product of two vectors on the plane.

▸ We have one: <1,-1,2> ▸ Form a second by

connecting two points

▸ <-2,3,-2>

A(1,4,2)

L

(3,1,4) <1,-1,2>

n

• n = 1, 1, 2 2, 3, 2 =
• ˆ

i ˆ j ˆ k 1 1 2 2 3 2

• = 2 6, 4 + 2, 3 2

= 4, 2, 1

MATH 200

▸ Answer: ▸ Point-normal (using A): -4(x-1) - 2(y-4) + (z-2) = 0 ▸ Standard: -4x - 2y + z = -10

A(1,4,2)

L

(3,1,4) <1,-1,2>

n = <-4,-2,1>

MATH 200

EXAMPLE 3

▸ Find the line of intersection of

the planes P1:x-y+z=4 and P2:2x+4x=10

▸ Start with a diagram: ▸ P1 has normal n1=<1,-1,1> ▸ P2 has normal n2=<2,0,4>