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jessicajx22

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I have no idea how to do it, all help will be appreciated.

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In summary, two planes, plane 1 and plane 2, intersect in the line with symmetric equation (x-1)/2 = (y-2)/3 = (z+4)/1. Plane 1 contains the point A(2,1,1) and plane 2 contains the point B(1,2,-1). To find the scalar equations of planes plane 1 and plane 2, we need to determine 3 points in each plane. Using the symmetric equation, we can find 2 additional points in plane 1, which are (x,y,z)=(1,2,-4) and (3,5,-3). A scalar equation of a plane can be found based on 3 points

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jessicajx22

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I have no idea how to do it, all help will be appreciated.

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I like Serena

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jessicajx22 said:

I have no idea how to do it, all help will be appreciated.

Hi Jessica, welcome to MHB!

A plane can be determined by 3 points that are in the plane.

So let's see if we can find 3 such points.

Obviously plane 1 contains point A(2,1,1).

So we need to use (x-1)/2 = (y-2)/3 = (z+4)/1 to find 2 more points.

Suppose each of them is 0. Then we must have x=1, y=2, z=-4. That is because for instance (1-1)/2=0.

Alternatively, if each of them is 1, then we must have x=3, y=5, z=-3, don't we?

Now we have 3 points in plane 1.

Do you already know what a scalar equation of a plane is?

And perhaps how to find it based on 3 points?

A plane is a two-dimensional flat surface that extends infinitely in all directions. A scalar equation, on the other hand, is a mathematical representation of a plane using variables and constants.

To determine if two planes intersect, you can set their scalar equations equal to each other and solve for the variables. If the resulting solution is a consistent set of values, then the planes intersect at a single point. If the solution is an inconsistent set of values, then the planes are parallel and do not intersect. If the solution is a dependent set of values, then the planes are coincident and intersect at infinitely many points.

No, two planes can only intersect at one point, be parallel, or be coincident. This is because a plane is a two-dimensional surface and can only intersect another plane at a single point.

To find the point of intersection between two planes, you can solve the system of equations formed by setting their scalar equations equal to each other. The resulting solution will give you the coordinates of the point of intersection.

Yes, planes can intersect at a 90-degree angle, but this is not always the case. The angle of intersection between two planes depends on the orientation of the planes and the direction of their normal vectors. If the normal vectors of the planes are perpendicular, then the planes will intersect at a 90-degree angle.

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