Question

The general equation of a plane is Ax + By + Cz = D, where A, B, C, and D are real numbers and A is nonnegative. Find the equation of the plane containing the points (2, 0, 0), (0, 4, 0), and (0, 0, 7). Show each step of your process. Then graph the plane.

Answer 1

@zepdrix

Answer 2

@Nnesha

Answer 3

@jim_thompson5910

Answer 4

@mathmale

Answer 5

What topics has your teacher gone over so far?

Have you learned about vectors and cross products?

Answer 6

I don't remember learning about that, but this is actually an assignment i didn't do from a few months ago. I'm doing it now to get a few extra points. I forgot most of the information, so that's why I need help. So it is possible but it doesn't ring a bell.

Answer 7

ok one moment

Answer 8

if we had something like (2, 0, 0) then this means

(x,y,z) = (2, 0, 0)
and

x = 2
y = 0
z = 0

do you agree?

Answer 9

yes that makes sence

Answer 10

Let's plug in (x,y,z) = (2, 0, 0)
Ax + By + Cz = D
A*2 + B*0 + C*0 = D
2A = D
D = 2A

Answer 11

What do you get when you follow those same basic steps, but plug in (x,y,z) = (0,4,0) ?

Answer 12

Ax + By + Cz = D
A*0 + B*2 + C*0 = D
D= 0A??

Answer 13

y = 4, not y = 2

Answer 14

OK

Answer 15

so hopefully you can see how D = 4B ?

Answer 16

yes i plugged in the wrong number

Answer 17

ok what happens when you plug in (x,y,z) = (0,0,7) ?

Answer 18

Ax + By + Cz = D
A*0 + B*0 + C*7 = D
D = C7?

Answer 19

or
D = 7C

Answer 20

So to recap, so far we know...

D = 2A
D = 4B
D = 7C

agreed?

Answer 21

yep :)

Answer 22

D is equal to 2A, 4B, 7C

there are infinitely many values for D. The smallest whole number D can be is the LCM of 2,4,7

what is the lowest common multiple of 2,4, and 7?

Answer 23

it would be 2^1 2^2 and 7^1 so 28?

Answer 24

yep, so D = 28 which allows us to find the values of A,B,C

Answer 25

ok cool

Answer 26

so now would have 28 is equal to 2A, 4B, 7C. right?

Answer 27

if 28 = 2A, then A = ???

Answer 28

umm would you divide however many 2's go into 28?

Answer 29

yes divide both sides by 2

Answer 30

A=14

Answer 31

yep, how about B?

Answer 32

7

Answer 33

and finally C?

Answer 34

4

Answer 35

summary

A = 14
B = 7
C = 4
D = 28

making the equation to be

Ax + By + Cz = D
14x + 7y + 4z = 28

Answer 36

ok awesome! So now how do i graph it?

Answer 37

are you allowed to use technology like a graphing program?

Answer 38

Yeah i think that's fine

Answer 39

first solve `14x + 7y + 4z = 28` for z to get

14x + 7y + 4z = 28
4z = 28-14x-7y
z = (28-14x-7y)/4

now onto the graph
there are many options to go with. Google is one good choice. If you were to google `z = (28-14x-7y)/4` it will show you what the graph looks like

http://prntscr.com/b3tzd0

the graph is interactive which means you can rotate things around at any angle

Answer 40

The only issue is that the x,y, and z axis aren't labeled too well.

Answer 41

you can also just graph by using the x, y, and z intercepts

plot the 3 points, then draw a plane through them all

Answer 42

Ok i will give it a try

Answer 43

This is what i was given to graph on

Answer 44

ok so you'll plot the three points like so. See attached

Answer 45

Ok thank you so much for all your help!!

Answer 46

imagine a flat plane going through all the points

the plane is shown as the purple triangle. Imagine it extended out infinitely in all directions along the flat surface (which is tilted up)

Answer 47

if I drew the plane with each side extended out infinitely, then it would fill up the entire space making it look non-tilted, but it is tilted

the interactive plotter shows what I mean

Answer 48

ok that makes sense