Question

Calculus III: Lines, planes, and surfaces in space.

Answer 1

@B25 @khoala4pham Find the standard and general form of the plane passing A(1, 1, 1,), B(4, 0, -1), and C(0, 3, -1).

Answer 2

Could you give me your definition of the standard and general form of a plane?

Answer 3

Ax + By + Cz = D

This is the standard form?

Answer 4

start with vector AB x vector AC

Answer 5

Plane equation (more complete)\[Ax+By+Cz=D\]\[n_v=<A,\ B,\ C>\]

Answer 6

Yes, like @B25 says, that cross product generates the normal to the plane. Can you proceed further given this knowledge?--Now you are able to reduce it back to the normal-point form of a plane.

Answer 7

Scalar Equation of a Plane\[a(x-x_0)+b(y-y_0)+c(z-z_0)=0\]\[P_0=(x_0,\ y_0,\ z_0)\]\[n_{vector}=<a,\ b,\ c>\]

Answer 8

\[V_{AB}=<3,\ -1,\ -2>\]\[V_{AC}=<-1,\ 2,\ -2>\]Evaluating \[V_{AB}\times V_{AC}\]

Answer 9

Just a little bit more for clarity, since AB and AC lie in the same plane, we need the plane's normal. Geometrically, the cross product of two vectors is another vector that is normal to both which means it is normal to the plane.

Answer 10

Do you need to know how to compute the cross?

Answer 11

No

Answer 12

I got <6,8,5>

Answer 13

@khoala4pham Thank you for the clarification that we are finding a normal vector; this normal vector will help us construct our plane equation. For my evaluation of the normal vector, we obtained\[V_{AB}\times V_{AC}=<6,\ 8,\ 5>\]

Answer 14

3 same answers can't be wrong...I got it too

Answer 15

Plane equation\[6(x-1)+8(y-1)+5(z-1)=0\]

Answer 16

Using point A(1, 1, 1), we get planar equation 6(x-1) + 8(y-1) + 5(z-1) = 0\[6x-6+8y-8+5z-5=0\]\[6x+8y+5z=19\]

Answer 17

@stamp , if you notice, if you substitute any of those other 3 points into your scalar equation, it gives the exact same equation. D is still 19 so the answer is consistent.

Answer 18

other 2 points** what I meant is that all 3 points yield the same plane (which we expect.)

Answer 19

@khoala4pham We just tested that, I did it using point B and @B25 did point C, we both got the same Ax + By + Cz = D

Answer 20

I am going to close this question, and instead of evaluating a similar problem, we are going to go onto finding the angle and line of intersection between two planes. Again, thank you @khoala4pham for your time and guidance. Already this is coming along quite nicely thanks to you.