Intersection Definition and 712 Threads

In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the intersection of objects is that which belongs to all of them. For example, in Euclidean geometry, when two lines in a plane are not parallel, their intersection is the point at which they meet. More generally, in set theory the intersection of sets is defined to be the set of elements which belong to all of them. Unlike the Euclidean definition, this does not presume that the objects under consideration lie in a common space.
Intersection is one of the basic concepts of geometry. An intersection can have various geometric shapes, but a point is the most common in a plane geometry. Incidence geometry defines an intersection (usually, of flats) as an object of lower dimension that is incident to each of original objects. In this approach an intersection can be sometimes undefined, such as for parallel lines. In both cases the concept of intersection relies on logical conjunction. Algebraic geometry defines intersections in its own way with intersection theory.

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  1. S

    Proofs of subspaces in R^n (intersection, sums, etc.)

    Homework Statement Let E and F be two subspaces of R^n. Prove the following statements: (n means "intersection") If EnF = {0}, {u1, u2, ..., uk} is a linearly independent set of vectors of E and {v1, v2,...vk} is a linearly independent set of vectors Note: Above zero denotes the...
  2. M

    Finding the Intersection of Two Circles: A Challenge

    hi everyone ! we have two circles that doesn't have intersections now we want to find a point on each circle that the distance of this two points are 'k' please help me . . .
  3. V

    Finding the volume of a 3-dimensional spherical intersection

    If two spheres of radius 1 intersect each other so the surface of each sphere passes through the other’s center how could you find the exact volume of the intersection? It doesn't matter what kind of method, it could be double or triple integration, geometry, polar, cylindrical or spherical...
  4. I

    Finding the point of intersection between two vectors

    Homework Statement L1 passes through (1,-4,0) and (9,0,4) L2 passes through (2,-3,-1) and (4,-3,3) Do L1 and L2 intersect? If so, where? Homework Equations Parametric equations(?) The Attempt at a Solution A = (1,-4,0) B = (9,0,4) C = (2,-3,-1) D = (4,-3,3) AB =...
  5. P

    Parametrics - intersection and collision

    Homework Statement x1t= 2-cos(pi*t) y1t= 3+7sin(pi*t) x2t= 3t+2 y2t= -(7/15)(3t+1)2 + 157/15 Find points of intersection and collision Homework Equations above? The Attempt at a Solution Well, to find the intersection I think I need to eliminate the parameters for both...
  6. T

    Intersection of probabilities.

    Hello, I need help with this problem: A: 67,000 Purchasing managers that are male B: 33,000 purchasing managers that are female C: 245,000 financial managers that are male D: 150,000 financial managers that are female Out of these 495.000 individuals , what is the probability that a...
  7. A

    Point and Circle intersection in 3 D

    Hi I need some support regarding a problem. I know the poistion of a point in 3D let say (sp,yp,zp) and I know the circle with Center (xc,yc,zc) having radius rc. My question is how to find the intersection point of the circle and a line in 3D. I know that we can find the POI in 2 D by...
  8. D

    Set notation (union and intersection)

    Homework Statement Find simpler notation for the two sets: A= \bigcup^{\infty}_{j=0}[j,j+1] and B= \bigcap_{j \in Z}(R minus\ (j,j+1)) Homework Equations The Attempt at a Solution Not really sure what it means by "simpler notation"... Does A=R since the union of...
  9. R

    Find some vector function whose image is the intersection of two surfaces

    Hi all, I'm quite new here, but it's been a while since I've been browsing through these forums for past answered questions for calculus and physics, but now comes the time where I'm the one needing help that's not been questioned yet. Homework Statement Find some* vector funcion r with...
  10. V

    Maple Approximating Line Intersection Equation with Maple

    I'm having a hard time creating a tidy equation to solve this seemingly simple problem. I have two points in standard 3D cordinate space, we'll call them C and D to be consistent with my work. The location of these points is given by (Ci,Cj,Ck) and (Di,Dj,Dk). I then have a third point, A...
  11. L

    Intersection of 2 subgroups is a subgroup?

    Homework Statement H and K are subgroups of G. Prove that H\capK is also a subgroup. The Attempt at a Solution For H and K to be subgroups, they both must contain G's identity. Therefore, e \in H\capK. Therefore, H\capK is, at least, a trivial subgroup of G. This was a test...
  12. C

    Lagrange Multipler and Max/Min point of intersection

    Homework Statement The plane 4x − 3y + 8z = 5 intersects the cone z^2 = x^2 + y^2 in an ellipse. Use LaGrange Multipliers to find the highest and lowest points on the ellipse. Homework Equations Lagrange Multiplier The Attempt at a Solution I guess I lack an understanding of...
  13. F

    Parametrization - circle defined by plane intersection sphere

    Show that the circle that is in the intersection of the plane x+y+z=0 and the sphere x2+y2+z2=1 can be expressed as: x(\vartheta) = (cos(\vartheta)-(3)1/2sin(\vartheta)) / (61/2)y(\vartheta) = (cos(\vartheta)+(3)1/2sin(\vartheta)) / (61/2)z(\vartheta) = -(2cos(\vartheta)) / (61/2) I'm really...
  14. C

    Proving complement of unions equals intersection of complements.

    Homework Statement Generalize to obtain (C1 U C2 U...U Ck)' = C1' intersect C2' intersect...intersect Ck' ' = complement Say that C1, C2,...,Ck are independent events that have respective probabilities p1, p2, ..., pk. Argue that the probability of at least one of C1, C2,...,Ck is equal to 1...
  15. D

    Intersection of a parabola with another curve

    Homework Statement For a any parabola with the equation y=kx^{2} I'm trying to find a curve that intersect every point of the parabola at right angles. Homework Equations For a perpendicular intersection the slope is -\frac{1}{m}The Attempt at a Solution I took the derivative and then took...
  16. R

    Find Intersection of Plane & Line, Does Line Lie in Plane?

    Homework Statement Find the point(s) of intersection (if any) of the plane and the line. Also determine whether the line lies in the plane. 2x-2y+z=12, x-\frac{1}{2}=-y-\frac{3}{2}=\frac{x+1}{2} Homework Equations 2x-2y+z=12 x-\frac{1}{2}=-y-\frac{3}{2}=\frac{x+1}{2} The...
  17. K

    Infinite union & infinite intersection

    I don't quite understand the meaning of "infinite union" and "infinite intersection". Is an infinite union ∞ U Ak k=1 being defined as a limit lim (A1 U A2 U ... U An) ? n->∞ How about an infinite intersection? Thanks!
  18. P

    Plot both sets and I want to highlight the intersection of A and B.

    I've two problems: Given are the two sets A = \left \lbrace (x_{0}, x_{1}, x_{2}, x_{3}) \in \mathbb{R}^{4} \mid x_{0}^{2} = \vec{x} \, ^{2}, x_{0} \geq 0 \right \rbrace and B = \left \lbrace (x_{0}, x_{1}, x_{2}, x_{3}) \in \mathbb{R}^{4} \mid (k_{0} - x_{0})^{2} = (\vec{k} -...
  19. M

    Plane through point and intersection of 2 other planes

    Homework Statement Find the plane that passes through the point (-1,2,1) and contains the line of intersection of the planes: x+y+z=2 2x-y+3z=1 The Attempt at a Solution First, I know that I need to find the line of intersection of the 2 planes. To do this, I used the cross product...
  20. S

    Intersection of a family of sets

    Homework Statement There is only a small issue that i am confused about... If we have a set \left(-\frac{1}{n},\frac{1}{n}\right), where n is a natural number. If we want to find the intersection of all such sets, my question is whether the result will be the set containing only...
  21. F

    Quick shpere intersection question?

    quick shpere intersection question?? Homework Statement Use the given info to answer the following questions a. find an equation of the sphere with the give center and radius. center (1,-11,3) radius 5 answer is (x-1)^2 + (y+11)^2 + (z-3)^2 = 25 where I am stuck is the next...
  22. U

    Modeling a lense using the intersection of two spheres

    Hi everybody! I am trying to model a lense shape using two circles of radii R and r, with one at the origin and the other offset upwards vertically by distance D. D must be less than R + r, but it must be greater than the larger of R or r. Thus, I have two equations, and the region of their...
  23. R

    How do i get the point of intersection of a line and a circle?

    How do i get the point of intersection of a line and a circle.. I got lots of information on this topic, but my requirement is not matching.. I got a line whose one end point lies at the origin of the circle.. and other end lies somewhere outside the circle.. Now i need the point of...
  24. I

    Volume of intersection of spheres

    Homework Statement Find the volume of the intersection of two spheres of radius 2, give that the center of each sphere lies on the surface of the other. The Attempt at a Solution I was trying to do this problem with volumes of revolution. I drew two circles, one with a center at -1, the...
  25. M

    The intersection of an empty collection of subsets of X is equal to X?

    Hi, I'm reading HL Royden's real analysis, though my question pertains more to set theory. Let X be a set. Then the intersection of an empty collection of subsets of X is equal to X. I understand this is not an intersection of empty subsets but it is still very counter-intuitive. Can...
  26. R

    Parametrize intersection of a plane and paraboloid

    Homework Statement Parametrize the intersection of the paraboloid z = x2 + y2 and the plane 3x -7y + z = 4 between 0 \leq t \geq 2*pi When t = 0, x will be greatest on the curve.Homework Equations The Attempt at a Solution I never really know how to do these kinds of problem. I am more...
  27. Saladsamurai

    Prove that the intersection of subspaces is subspace

    Homework Statement Prove that the intersection of any collection of subspaces of V is a subspace of V. Okay, so I had to look up on wiki what an intersection is. To my understanding, it is basically the 'place' where sets or spaces 'overlap.' I am not sure how to construct the problem...
  28. P

    How can I find the points of intersection between two cubes in 3D space?

    hi, I need to find the points of intersection between two cubes, and the algorithm should give the intersection points even though my cubes are rotated in any direction. i need to know, Is there any general algorithm for this and what is the current research(any method) in finding the...
  29. Y

    (A - B) union (B- A) = (A union B) - (A intersection B)

    Why is this true: (A - B) union (B- A) = (A union B) - (A intersection B) wouldn't the union of A and B everything that is in A or B? And since A - B and B - A don't contain any elements from the other set, wouldn't the union of these be equal to union of A and B? So wouldn't it make sense...
  30. M

    Determining the equation of an ellipse from its intersection with a parabola

    Homework Statement The vertex of the parabola y^2=2px is the center of an ellipse. The focus of the parabola is an end of one of the principle axes of the ellipse, and the parabola and ellipse intersect at right angles. Find the equation of the ellipse. Homework Equations...
  31. H

    Find the equation of the line of intersection of the planes:

    Homework Statement 2x-y-z=3 and x+2y+3z=7 Homework Equations The Attempt at a Solution Im stumped on this problem because initially i thought all i had to do was make z, or another variable zero and then just solve. However, it then turns into a nasty problem. Most of the...
  32. P

    How Do You Find the Equation of the Intersection of Two Planes?

    urgent, intersection of two planes Homework Statement Find the equation of the intersection between 2x + 3y - 4z = 12 and 5x + 2y + 3z = 7. Homework Equations none. I don't know how to use cross products, but is there another way? The Attempt at a Solution I don't know...
  33. C

    Volume of the intersection of two cylinders by polar co-ordinates

    Volume of the intersection of two cylinders by cylinderical co-ordinates Homework Statement find Volume of the intersection of two cylinders by cylindrical co-ordinates The Attempt at a Solution IN the attached file I found it's 8(a^3)/3 It should be 16 not 8
  34. R

    Finding the intersection point of two 3D vectors?

    Homework Statement **NOTE: One coordinate unit = 1000 feet. Also, the helicopter and the airplane are TWO SEPARATE moving vehicles.** At noon (12:00 PM), a helicopter is observed from point A (7, 0, 0) in the direction of vector -4i + 2j + 5k, and simultaneously from point B (0, 4, 0.25) in...
  35. J

    Infinite union and intersection

    Homework Statement Given a set A \in R^m, B_n \in R^m for n \in N, show that A \ Union {from n = 1 to inf} B_n = Intersection {from n = 1 to inf} (A \ B_n} Homework Equations Same equation as above The Attempt at a Solution I think I have a solution in mind, but I wanted to...
  36. K

    Fortran Understanding Union & Intersection of Two Sets

    I have a program to define union and intersection on 2 sets. I know I need 2 different input files, but other than that, I'm clueless.
  37. J

    Parameterize the curve of intersection

    Homework Statement Parameterize the curve of intersection of the cylinder x^2 + y^2 = 16 and the plane x + z = 5 Homework Equations The Attempt at a Solution i think i must first parameterize the plane x = 5t, y = 0, z = -5t then i think i plug those into the eq. of the...
  38. M

    Infinite sets, intersection, nested intervals.

    Homework Statement Let [a,b] be an interval and let A be a subset of [a,b]. and Suppose that A is an infinite set. Let z be the unique point that belongs to all of the intervals [an, bn]. Show that if I is any interval that contains z, then A intersect I is infinite. Homework...
  39. D

    Real World Application: Intersection Signal Sensors

    Hey everyone! I recently picked up a motorcycle for class/work commutes because walking home at 11pm from the lab is not fun. But I have come across something of interest. Some intersections keep a green light on the main street and red on cross streets until a car pulls up. Before...
  40. A

    Tricky subspace & intersection Problem

    Homework Statement I am trying to solve this problem: Let W_1, W_2, W_3 be subspaces of a vector space, V. Prove that W_1 ∩ (W_2 + ( W_1 ∩ W_3)) = (W_1 ∩ W_2) + (W_1 ∩ W_3). Can someone help me show this? I have tried using Dedekind's law, but not sure it that is the way to go. The...
  41. K

    Linkage analysis using intersection of three sphere

    Hello. I am creating an algorithm for finding the locations of nodes connected by rigid links. Some nodes in the system are constrained in place, and to one node (a *special* node), a displacement is applied. The algorithm should be able to determine the locations of the other nodes by...
  42. J

    Find the closest point to the origin on the curve of intersection to a cone

    Homework Statement find the point closest to the origin on the curve of intersection of the plane 2y+4z=5 and the cone z^2=4x^2+4y^2 Homework Equations The Attempt at a Solution see 40 attachement. I found the used f(x,y,z)=x^2+y^2+z^2 and found its gradient. found ggrad and...
  43. Z

    System of linear equation question: Intersection of three equations

    Homework Statement Solve the system of the linear equations and interpret your solution geometrically: 2x + y + 2z - 4 = 0 [1] x - y - z - 2 = 0 [2] x + 2y -6z - 12 = 0 [3] The Attempt at a Solution I've tried to eliminate the y variable: [1] + [2] 3x + z - 6 = 0 [4]...
  44. R

    Point of intersection for sine and cosine functions

    The problem is finding the points of intersection for two given functions. f1=sin(-\pi*x) f2=1+cos(-\pi*x) I've plotted the functions using Maple. http://dl.getdropbox.com/u/12485/plot.png And I'm quite certain that to find the points of intersection, I have to set f1=f2 which...
  45. C

    Linear algebra, point of intersection

    Homework Statement 1) x=3+t y=2-4t z=-5+11t 2)12x+10y-4z Find the point where these two lines intersect. please help!
  46. T

    Polynomial intersection question

    V1={cx^3+ax^2|c,a exists in R} V2={dx^3-bx^2 -d|d,b exists in R} find all the values of a,b that f(x)\epsilon V1\cap V2 ?? find all the values of a,b that f(x)\epsilon V1+ V2 ?? i know how to solve such question using vectors but they are not using using vector but some hoe check the...
  47. S

    Nested sequence of compact sets in Rn has a non-empty intersection?

    There's a theorem that says any nested sequence of compact sets in Rn always has a non-empty intersection. So there is something wrong with this counterexample. I'm not able to see what's wrong: Consider the interval Un = [2-1/n, 1+1/n] for n=1, 2 and 3. Isn't the intersection of U1, U2 and...
  48. E

    Proving the Intersection of Open Sets is Open

    we can prove that: the intersection of a finite number of open sets is open. how about: the intersection of any number of open sets? it's maybe not open.but how to prove it or the example? :smile:
  49. J

    Finding points of intersection using vectors

    Find the points of intersection of the lines... L1: R = 2i + 3j + 3k + t(i - 2j + 5k) L2: (x + 3) / 2 = (y + 1) / 2 = -z (I assume the plural in points is wrong... since that would be impossible) R, i, j, k are vectors; x, y, z are not --- x = 2 + t y = 3 - 2t z = 3 + 5t for some value(s) of...
  50. W

    Intersection Form in 4-D. Followup.

    : I hope this is not too dumb. I am kind of unclear on some issues: i) I understand that every bilinear map over a fin. dim. V space /F has a matrix representation (depending on the choice of basis ). Still, the intersection form ( in an orientable 4-mfld M.) is a map...
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