Vectors, Planes, and intersections

In summary, the three planes intersect at point (62,64,100), which is different from the point given in the homework statement.
  • #1
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Homework Statement



The planes 3x+2y+z=6 and x+2y+5z=1- intersect along the line (x+2)/2 = (y-6)/(-7/2)= z.

A third plane passes through the origin and is perpendicular to the intersection of the first two planes, at what point do the three planes intersect?

Homework Equations



Intersection in parametric eq:

x = -2 + 2t
y = 6 - 7/2t
z = t

The Attempt at a Solution



At first glance I knew that a normal vector will be perpendicular to the plane, so I just took the vector from the intersection <2, -7/2, 1> and used that as the normal vector for the third plane. I knew it passes through origin so i used point (0,0,0)

I get

2x - 7/2y + z = D

I input (0,0,0) for (x,y,z) and get D= 0

Thus the equation for the third plane is

2x - 7/2y + z = 0


At this point I plot everything in maple and the intersection that was given to me doesn't even look like it's the true intersection, none the less my third plane doesn't look like its perpendicular to my intersection either.

We have never done 3 planes intersecting but I assume it similar to 2 planes, where I set a variable equal to zero such as x=0 and then solve the system of equations. Once the other 2 are found, we can then just find the third and that will be the point in the intersection. I am not sure how to find the vector
 
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  • #2
ParoXsitiC said:

Homework Statement



The planes 3x+2y+z=6 and x+2y+5z=1- intersect along the line (x+2)/2 = (y-6)/(-7/2)= z.

A third plane passes through the origin and is perpendicular to the intersection of the first two planes, at what point do the three planes intersect?

Homework Equations



Intersection in parametric eq:

x = -2 + 2t
y = 6 - 7/2t
z = t

The Attempt at a Solution



At first glance I knew that a normal vector will be perpendicular to the plane, so I just took the vector from the intersection <2, -7/2, 1> and used that as the normal vector for the third plane. I knew it passes through origin so i used point (0,0,0)

I get

2x - 7/2y + z = D

I input (0,0,0) for (x,y,z) and get D= 0

Thus the equation for the third plane is

2x - 7/2y + z = 0


At this point I plot everything in maple and the intersection that was given to me doesn't even look like it's the true intersection, none the less my third plane doesn't look like its perpendicular to my intersection either.

That might be caused by not using the "scaling=constrained" option in the Maple plot.

We have never done 3 planes intersecting but I assume it similar to 2 planes, where I set a variable equal to zero such as x=0 and then solve the system of equations. Once the other 2 are found, we can then just find the third and that will be the point in the intersection. I am not sure how to find the vector

I would multiply the equation of the third plane by 2 on both sides to get rid of the fractions. To find the intersection point you could either:

1. Solve the three plane equations in three unknowns for ##x,y,z## using substitution, row reduction, or determinants.

2. Probably easier, substitute the parametric equation of the original line of intersection into the third plane, solve for ##t##, and use that ##t## to get ##x,y,z##. Then check it trying ##(x,y,z)## in each plane.
 
  • #3
x+2y+5z=1- should be x+2y+5z=10

Also I had make sure scaling was constrained, but now trying to do it in maple it looks right. so must of been a small mistake.

None the less taking your advice I got

t = 100/69

solving for xyz:

x = 62/69
y = 64/69
z = 100/69

And it checks for all 3 planes, so I guess it's right :)
 
Last edited:
  • #4
ParoXsitiC said:
x = 62/69
y = 64/69
z = 100/69

And it checks for all 3 planes, so I guess it's right :)

That's what I would guess too. :cool:
 

What is the definition of a vector?

A vector is a mathematical object that has both magnitude and direction. It is commonly represented by an arrow pointing in the direction of its magnitude.

How do you calculate the dot product of two vectors?

The dot product of two vectors is calculated by multiplying the magnitudes of the vectors and then multiplying the cosine of the angle between them.

What is the equation of a plane in three-dimensional space?

The equation of a plane in three-dimensional space is Ax + By + Cz + D = 0, where A, B, and C are the coefficients of the x, y, and z variables respectively, and D is a constant.

How do you find the intersection of two planes?

To find the intersection of two planes, you can set their equations equal to each other and solve for the values of x, y, and z that satisfy both equations.

What is the significance of the cross product of two vectors?

The cross product of two vectors is a third vector that is perpendicular to both of the original vectors. It is often used in physics and engineering to calculate torque and determine the direction of rotation.

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