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. R

    Finding Area of Circle and Cardoid Intersection

    Homework Statement Find the area of the region that is inside the circle r = 6cos(theta) but outside the cardoid r = 2 + 2cos(theta) Homework Equations r = 6·cosθ r = 2 + 2·cosθ The Attempt at a Solution intersections of the two curves. 6·cosθ = 2 + 2·cosθ → 4·cosθ = 2...
  2. Q

    Intersection of two 3D parametric curves

    Hi, I have two parametric curves defined in three dimensions, which are functions of a variable t, like so: x1 = f1(t) y1 = f2(t) z1 = f3(t) x2 = f4(t) y2 = f5(t) z2 = f6(t) I am trying to find the intersection of these two curves, but I am having some difficulty with the...
  3. B

    Finding Parametric Equation of Tangent Line to Intersection of Surfaces?

    Homework Statement Hi, I need help with the following. I'm asked to find the parametric equation of the tangent line to the curve of the interrsection of the paraboloid z = x^2 + y^2 and the ellipsoid 4x^2+y^2+z^2 = 9 at the point (-1,1,2). Homework Equations I think I'm asked to find...
  4. P

    Orbit Intersection Question (classical mechanics)

    Homework Statement A comet is going in a parabolic orbit lying in the plane of Earth's Orbit. Regarding Earth's orbit as circular of radius "a," show that the points where the comet intersects Earth's orbit are given by: cos(theta)= -1 + (2*p)/a where p is the perihelion distance of the...
  5. P

    Labeling Intersection in Venn Diagrams

    just a clarification cause it's been so long if you have intersection in a venn diagram, is it general practice to label the section including the intersecting diagrams. say you have two sets of 10 with 5 things in the intersect. do you label it 10 in the 2 sets and 5 in the intersect. or. 5...
  6. A

    Contour map and Intersection points

    Homework Statement (i) Drawing a contour map for the function h(x.y) = -12-4x^2+16x-y^2-8y (ii) (Continuing from i) at the point (1,-1,7) which direction to move to have the maximum increase in height? (iii) Find the point closest to the origin on the curve of intersection of the...
  7. C

    What Is the Necessary Condition for Three Planes to Intersect Along a Line?

    Homework Statement Find a necessary condition for the three planes given below to have a line of intersection. -x +ay+bz=0 ax-y+cz=0 bx+cy-z=0 Homework Equations in order to get a line of intersection between the planes..i know i need one line of the matrix to be [0 0 0|0]...
  8. P

    Finding Intersection of Line and Circle: A Challenging Problem

    Homework Statement I can't find the intersection for Line y = 3/4x - 35/4 and Circle y^2 + x^2 = 25 Homework Equations The Attempt at a Solution
  9. C

    Points on the curve of intersection

    hi i need to find the points on the curve of intersection of x^2+(y-1)^2+(z-1)^2 = 1 and 2x + y + 2z = 4 which are nearest and furthest furthest from the origin. Also the min and max distances. I'm not looking for you guys to do this four me, I'm kind of lost and don't know where to go. I...
  10. P

    Intersection of a sphere and plane problem

    Hello, I was wondering if anyone could offer some advice on this one. I have a multi-part problem, in which I can't get the second part. It starts like this: (a) Find the equation of the sphere passes through the point (6,-2,3) and has a center of (-1,2,1). So I did this find and...
  11. P

    Find the line of intersection between 2 Planes

    Hello, I am trying to find the line of intersection between these two planes: P_1 = x + 2y -9z = 7 P_2 = 2x - 3y + 17z = 0 I found the direction vector needed for the line of intersection between these two points by taking the cross product of the P_1 normal vector and the P_2 normal...
  12. P

    Quick Vector intersection question

    So, I am to find a parametric equation for the line through P_1 and P_2 and then determine (if possible) the points at which the line intersects each of the coordinate planes. P_1 (5,-2,4) P_2 (2,6,1) \vec{P_{1}P_{2}} <-3,8,-3> x = 5-3t y = -2+8t z = 4-3t So I got this far, but...
  13. I

    Two cars approaching an intersection

    Hi, I have this problem that I can't seem to figure out. I'm taking physics in French but I will try my best to translate it. At an instant, two cars, A and B, are 10km away from an intersection of two perpendicular roads. Car A is moving towards the East and has a speed of 30km/h whereas...
  14. Y

    Curve of intersection of plane and surface

    this is the problem and I've worked for mostof it, but I am not sure I've got the right way to go about it, or if the ansewr i have found it right. here, I am using the method for finding the vector equation given the plane 2x + 6y + z = 6 intersects the paraboloid z = x^2 + y^2, find and...
  15. M

    Proving No Intersection: y=2x-1 & y=x^4+3x^2+2x

    Heres the quesion Prove that the line whose equation is y=2x-1 does not intersect the curve with equation y=x^4 + 3x^2 +2x. We are suppose to solve this using indirect proof, thus assuming the equations do intersect, and proving that wrong. i let the y's equal each other, but that isn't...
  16. D

    Intersection of cosets is empty or a coset

    "Let H and K be subgroups of a group G. Prove that the intersection xH\cap yK of two cosets of H and K is either empty or is a coset of the subgroup H\cap K." I'm stuck here.
  17. C

    Solving Two Cars Intersection Problem: Velocity of Car B Relative to Car A

    These two problems are giving me some guff. Any ideas on how to solve? Two cars approach an intersection of perpendicular streets. Car A is moving north at 10.5 meters per second, and Car B is moving west at 17.1 meters per second. What is the velocity of Car B relative to Car A? (In other...
  18. Z

    Find a formula for the intersection math

    Let I=(a,b) and J=(c,d) with I and J having a nonempty intersection. Find a formula for the intersection of I and J and prove it. When a<c I have found that the intersection is [c,b]. Now I need to prove that it is. The way I intend to prove it is by showing that [c,b] and the intersection...
  19. U

    Show that nZ intersection mZ= lZ

    hello i have two questions and i need answers for them first one: in the additive group (Z,+) show that nZ intersection mZ= lZ , where l is the least common multiple of m and n. The second question is : Given H and K two subgroups of a group G , show the following: (H...
  20. T

    Finding the Intersection of Subspaces with Given Spanning Vectors

    How do you find the intersection of subspaces when the subspaces are given by the span of 3 vectors? For example, U is spanned by { X1 , X2 , X3} and V is spanned by { Y1, Y2, Y3}. Thanks in advance.
  21. J

    Calculating the Intersection of Two Stones Dropped from a Cliff

    A stone is droped from the top of a cliff of height 256 feet, and, at the same instant, another stone is projected vertically upwards from the ground with a speed of 96 ft/s. Find where they will meet? I have the following: v1=u1+at, v2=u2+at, s1=u1t+1/2at^2, s2=u2+1/2at^2, I also have...
  22. S

    Intersection of a paraboloid and a plane

    Question: Consider the intersection of the paraboloid z = x^2 + y^2 with the plane x - 2y = 0. Find a parametrization of the curve of intersection and verify that it lies in each surface. How I went about it: x = 2y z = (2y)^2 + y^2 = 5y^2 Set y = t, then x = 2t y = t z = 5t^2...
  23. S

    Intersection of a plane and cylinder

    Question: Find a parametric representation for the curve resulting from the intersection of the plane 3x + y + z = 1 and the cylinder x^2 + 2y^2 = 1. What I did: x = cost y = sint/sqrt(2) z = -3cost - (sint/sqrt(2)) + 1 I think I'm doing this correctly but the answer seems too easy...
  24. T

    Finding the Point of Intersection: Y=x+6 and Y=x^3

    may i know how to find the point of intersection between y=x+6 and y=x^3 ??
  25. M

    Intersection of spherical shell and a plane

    I need to know the area of the intersection between a sperical shell and a plane in spherical coordinates. By "shell" I mean a sphere with some differential thickness dR. Basically, I know that the intersection of a sphere and a plane is a circle. But I want to consider this sphere having...
  26. S

    Sphere and cylinder intersection

    Hi, Recently i tried (and failed) to calculate the intersection volume of a sphere and a cylinder. I found this simple problem seems not so simple for me. Searching on the web, nothing on that, so if someone can help me thank you. (the simplified solution with the intersection area of...
  27. T

    Intersection of Two Lines

    I have tried but doesn't work out well... Find the point at which the normal through the point (3,-4) to the line 10x+4y-101=0 intersects the line. Originaly I thought that it should be found by doing the dot product of: (x-3,y+4)dot(-4/10,-10/4)= 0
  28. D

    Intersection of cylinder and plane

    I'm trying to find the parameterization of the intersection of a cylinder x^2+y^2=1 and the plane x+y+z=1, but I'm not exactly sure how to go about it. Any guidance on how to find this intersection in a parameterized form would be most appreciated. In general I don't know a great deal about...
  29. A

    Does the Finite Intersection Property Allow Multiple Points in T1 Spaces?

    Let X be a space. Let \mathcal{D} be a collection of subsets of X that is maximal with respect to the finite intersection property. Show that if X satisfies the T_1 axiom, there is at most one point belonging to: I = \bigcap _{D\in \mathcal{D}}\bar{D} A collection of subsets has the...
  30. E

    How do you solve for the intersection of a line and a cosine function?

    Suppose you are given the equation of a line, and a given cosine function that the line intersects. How do you solve algebraically, that is non-graphically, for the point of intersection of the line and the cosine function? Inquisitively, Edwin
  31. M

    Intersection of line and circle

    I'm trying to find the points of intersection of line and circle with equations: (x-p)^2 + (y-q)^2 = r^2 (y-y1)*(x2-x1)-(x-x1)*(y2-y1)=0 but i can't handle with this. Can anyone help me?
  32. C

    Finite intersection of open sets is open

    In my multivariable calc class, we're asked to prove that the finite intersection of open sets is open. I've tried to find help on the internet but couldn't find anything to help. I understand somewhat the idea of "nesting sets" that some proofs use .. can anyone help me understand this to prove...
  33. Q

    Curve of intersection between surfaces

    I looked through some books and couldn't find how to find curves of intersection between surfaces. My question asks: explain why the curvature between surfaces z=x^2 and x^2+y^2=4 is the same of intersection between the surfaces z=4-y^2 and x^2+y^2=4. please help i feel really dumb right...
  34. N

    Proving Equivalence Relations: Intersection of Two Sets

    Hi All I have a problem with Set theory. I am given to prove the following; Is the intersection of two equivalence relations itself an equivalance relation? If so , how would you characterize the equivalnce sets of the intersection? Regards, Nisha.
  35. S

    Find intersection point between a ray and a face?

    I have a 3D polygon and a ray! Please tell me how can I know if they intersect together and how to find intersection point between them? :confused:
  36. B

    Proving the Intersection of Subspaces is a Subspace: A Vector Space Proof

    Hi can someone please help me with the following question. Such questions always trouble me because I don't know where to start and/or cannot continue after starting. Q. Let H and K be subspaces of a vector space V. Prove that the intersection of K and H is a subspace of V. By the way...
  37. L

    What Is the Intersection of Subspaces U and V in R^3?

    I have 2 subspaces U and V of R^3 which U = {(a1, a2, a3) in R^3: a1 = 3(a2) and a3 = -a2} V = {(a1, a2, a3) in R^3: a1 - 4(a2) - a3 = 0} I used the information in U and substituted it into the equation in V and I got 0 = 0. So, does it mean that the intersection of U and V is the whole...
  38. P

    Solving x²=-1/2ln(x) with x in (0,2]

    Hi,:-p so, I can't solve the embarissing: x^{2} = -\frac{1}{2}ln(x) , where x \in ]0, 2] or (0 < x \geq 2 ) any hep would be nice... thanx for your pacience!
  39. Y

    Intersection points of the graph its solutions?

    The graphs of f(theta) = 2sin(theta) - 1 , and g(theta) = 3cos(theta)+2 are given. What equation would have the intersection points of the graph its solutions? ummm... what does this mean? and how do i solve it?
  40. S

    Intersection of a line and a plane, for what value(s) of k ?

    Well, From what I understand, to determine the intersection of a line and a plane, we use parametric form of the line and substitute the values of x, y and z into the Cartesian equation of the plane, correct? so, given the line x = 2 + 4t y = -1 + kt <=== note the 'k' variable z = 5...
  41. S

    Why must every homogeneous system of planes have at least 1 intersection point

    Well, I'm doing homework (again). I was introduced to homogeneous systems of planes and then asked why there must be at least 1 intersection point. The book gives very little (one sentence) on homogeneous systems so I tried to search around online. My guess is that since all of the...
  42. S

    Solve Intersection Points of f(x) & g(x) with Algebra

    Hi guys, I'm just wondering is it possible to solve the following using algebra to obtain the points of intersection of the two curves f(x) = 6sqrt(x) and g(x) = [(x+5)^2]/36 I got to the point where i reconized that the inverse of g(x) = 6sqrt(x) - 5 which looks a lot like the function...
  43. wolram

    Exploring the Intersection of Jury St, Banbury Rd, and River Avon

    I have jury st in front of me I have banbury Rd to my right I have the river avon behind me.
  44. Bob3141592

    Is the Intersection of Nested Subsets Always Non-Empty?

    I'm just an interested laymn, and I'm trying to improve my knowledge in some areas where I'm weak. To this end, I found that Shilov's Elementary Real and Complex Analysis was highly recommended, and the Dover edition was available for only ten bucks, so how could I go wrong? But it didn't take...
  45. mattmns

    Line of Intersection of two Planes

    How do I find the line of intersection of two planes? I have an idea, but both of the planes have a -2z ie. Plane 1: 10x-4y-2z=4 Plane 2: 14x+7y-2z If I set them both equal to each other, I lose the z part. So, is there some other way to solve this, or am I missing something? Thanks!
  46. F

    Verifying Vector Intersection: Two Questions

    I have two questions I need to make sure if I'm doing correctly. Its vectors. 1) For what values of a are the vectors i+3j-k and i+aj+k i) inclined at 30 degree angle cos@ = n1.n2 / |n1||n2| cos^2(30) * 11(2+a^2) = ((3a)^2) a=sqrt22 ii) perpendicular (1,3,-1).(1,a,1)=0...
  47. C

    Intersection of cyclic subgroups

    This time I need a yes/no answer (but a definitive one!): Suppose we have a group of finite order G, and two cyclic subgroups of G named H1 and H2. I know the intersection of H1 and H2 is also a subground of G, question is - is it also cyclic? And can I tell who is the creator of it, suppose I...
  48. S

    Volume of Intersection of a Cone with a Sphere

    Hey, im trying to write a program that computes Volume of Intersection of a Cone with a Sphere. Can anyone point me to the math i need to know. Any links, material is good. Thanx
  49. A

    Intersection coordinates are my points right?

    Hi can someone please check that my points of intersection are correct? The question was determine the coordinate of the intersection point of \frac {(x-3)^2} {9} + \frac {y+2)^2} {4} =1 and y=2x-3 I after putting the second equation into the first and then expanding and solving...
  50. K

    Using the Fourier transform on unordered sets to determine if any intersection

    Hi, this forum looks great and I'm glad to have found it. Now to my first question. Basically, I want to know if there is any literature on using the Fourier transform on unordered sets in order to see if two sets intersect (and how many times). I welcome any alternative approaches, esp. if...
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