What is Intersection: Definition and 711 Discussions

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. B

    Find the Vector equation of the curve of intersection (Calculus 3)

    Homework Statement Find the Vector equation of the curve of intersection of X^2+y^2+z^2=10 and x+y=4 using x=2+sin(t) for parameter. Homework Equations The Attempt at a Solution I know it is a sphere with a plane cutting across it, however I have no idea how to proceed from...
  2. M

    A set is closed iff it equals an intersection of closed sets

    Homework Statement Let M be a metric space, A a subset of M, x a point in M. Define the metric of x to A by d(x,A) = inf d(x,y), y in A For \epsilon>0, define the sets D(A,\epsilon) = {x in M : d(x,A)<\epsilon} N(A,\epsilon) = {x in M: d(x,A)\leq\epsilon} Show that A is...
  3. T

    2D Circle and rectangle intersection tests

    What I'm looking for is an algorithm to find the details on the intersection of of a circle and rectangle in two dimensional Euclidean space. The information I need to find is straightforward enough; all I need is to know whether the rectangle and circle are not intersecting, partially...
  4. S

    Find a Vector parallel to the line of intersection

    Homework Statement Find a vector parallel to the line of intersection of the planes given by the equations 2x-3y+5z=2 and 4x+y-3z=7. Homework Equations How do I go about this? I know we have two vectors <2,3,5> and <4,1,-3> but where do I go from here? The Attempt at a Solution...
  5. B

    Intersection of a line and circle

    Homework Statement The line x+y=2a-1 intersects the circle x2+y2=a2+3a-3 at point (m,n). When m*n reaches its minimum value, what is the value of a?Homework Equations Equation One x + y = 2a - 1 Equation Two x2 + y 2 = a 2 + 3a - 3
  6. I

    Finding the point of intersection and distance of a vector

    Homework Statement Let P be the plane of Problem 9 and let L be the line perpendicular to P and passing through the point p=(1,1,2). Find the point of intersection between P and L and use it to compute the distance from p to P. Question 9 is this: Use the cross product to obtain the normal...
  7. W

    Intersection of Planes in R^3 and Dense Subsets of R^3

    Hi, All: This is a post from another site that was interesting but was not answered: can I reasonably > argue that three planes in 3-space are not likely > to intersect at a point using the fact that >t GL(3,R); > the subset of invertible 3x3-matrices has measure 0 > in...
  8. A

    Intersection of sets with infinite number of elements

    I have to decide whether the following is true or false: If A1\supseteqA2\supseteqA3\supseteq...are all sets containing an infinite number of elements, then the intersection of those sets is infinite as well. I think I found a counterexample but I'm not sure the correct notation. I to...
  9. M

    A question about intersection of system of sets

    The book I am reading says that \bigcap \phi because every x belongs to A \in \phi(since there is no such A ) , so \bigcap S would have to be the set of all sets. now my question is why every x belongs to A \in \phi.In other word I don't completely understand what this statement mean. sorry if...
  10. P

    What are the solutions for the intersection of y=abs(x) and y=(x^2)-6?

    Homework Statement The graphs y=abs(x) and y=(x^2)-6 intersect at x=3 and x= -3 What is confusing me is when I set them equal to each other and solve (x^2)-x-6=0 and (x^2)+x-6=0 I get -3,+3,-2,+2 What is the deal with the negative 2 and pos 2? Homework Equations The...
  11. Y

    Find the points of intersection of the curves y=2sin(x-3) and y=-4x^2+2?

    Can someone do it without using a graphing calculator? The question specifically states not to use "Trace". I don't understand how to do it algebraically, and I'd love it if someone could teach me. Please and thanks!
  12. C

    Find the point of intersection of three planes

    Homework Statement The plane P1 contains the points A,B,C, which have position vectors a=(0,0,0), b=(1,1,8) and c=(0,1,5) respectively. Plane P2 passes through A and is orthogonal to the line BC, whilst plane P3 passes through B an is orthogonal to the line AC. Find the coordinates of r, the...
  13. M

    Bending moments diagram at T intersection

    Homework Statement Draw the axial force, shear force and bending moment diagrams. Show the locations and magnitude of the maximum and minimum values. Homework Equations See diagram. The Attempt at a Solution I've worked out the values from points A-B and then from points B-D. I...
  14. A

    Intersection of line and plane

    Intersection of line and plane! Homework Statement Intersection of line and plane! Okay i to find the common points of line and plane Question r=i+j+A(2k-j) and r . (i+j) = 4 Homework Equations I heard that it is easier to use the vector equation in the form r . n = p The...
  15. A

    HELP Intersection of two lines (VECTORS)

    HELP! Intersection of two lines (VECTORS) Homework Statement Find the common point of the lines r=i+j+k+x(j-3k) and r=i+y(k-j) Homework Equations The Attempt at a Solution If the lines intersect then there are numbers x and y such that i+j+k+x(j-3k)=i+y(k-j) The two...
  16. P

    Determining intersection depth of two polyhedra in a specific direction?

    I finally got my GJK algorithm working and now i want to be able to find the depth of the intersection of the two polyhedra in a particular direction (the direction of momentum). I figure the best way would be to find the distance from the origin of the minkowski difference to the hull of the...
  17. E

    Plotting an intersection seam on a contour plot ?

    Hi, I have two 2D functions (surfaces), s1(x,y) and s2(x,y) defined via Interpolation. They intersect forming an intersection seam (which is a line). I can plot both functions using Plot3D and I can also plot the seam on the same 3D plot by means of the MeshFunctions option. The comand I...
  18. B

    How does the intersection form change when changing coefficient rings?

    Hi, All: The intersection form ( , ): H_n(M,R)xH_n(M,R)-->Z ; Z the integers and R any coefficient ring, in a 2n-manifold is well-defined in homology, i.e., if (x,y)= c , and x~x' and y~y' , then (x',y')=c Still, how is the value of the intersection form affected by changes in...
  19. A

    Finding Intersection and Tangent Lines of Parametric Curves | Step-by-Step Guide

    I need to find the point of intersection of the curves x^2 + y^2 =1, z= 0 and x=cost, y=sint, z=t. I plugged in the latter equation into the former and got (1,0,0) as an answer but I'm not exactly sure why that works, I can't visualize how plugging in the parts of a parametric equation will...
  20. M

    Infinite intersection of indexed sets

    Every element of a set A can be written a=w.a_1a_2a_3\ldots{a_n}\ldots with w, a_n\in\mathbb{Z} and 0\leq a_n\leq9 for every n\in\mathbb{N}. If A is bounded, there exists a greatest whole part \overline{w} of the elements of A, and because any set S of elements a_n is bounded, for every n, there...
  21. M

    Infinite intersection of indexed sets

    Consider the set A_n=\{0.9, 0.99, 0.999,...\} , where the greatest element of A_n has n 9s in its decimal expansion. Then 0.999\ldots=1\in\bigcap_{n=1}^\infty{A_n}. Is this possible even though \not\exists{n}(1\in{A_n})? Edit: I see that 0.999\ldots=1\not\in\bigcap_{n=1}^\infty{A_n}...
  22. J

    Finding Points of Intersection for Polar Curves

    Homework Statement I need to find the 2 points of intersection (in polar form) of the two curves. I know just by looking that the origin will be one of the points, (0,0) The Attempt at a Solution I have approached this two different ways, 1. set them equal to each other and tried to...
  23. A

    Intersection of Plane x=y and Surface in R3

    What does a surface in R3 that intersects plane x = y at a line for every value of x represent? My first intuition is that it represents a plane because in R3 planes intersect at lines but I feel like there is a counterexample to this.
  24. W

    Is the Transversal Intersection of Manifolds a Manifold?

    Hi, All: Given manifolds M,N (both embedded in $R^n$, intersecting each other transversally, so that their intersection has dimension >=1 ( i.e. n -(Dim(M)-Dim(N)>1) is the intersection a manifold? Thanks.
  25. S

    Limit of the intersection of events

    Hi, I keep seeing this come up A1 ⊇ A2 ⊇ A3 ... is an infinite decreasing sequence of events. Prove from first principles that P(intersection of Ai from i=1 to infinity) = Lim P(An) as n--> infinity All i can think of is that since each is a subset of the preceding, then A1 ∩...
  26. D

    Parametric equation of an intersection.

    Homework Statement Find the parametric equations of the intersection line of two planes 2x - 3y - z + 1 = 0 and 3x - 2y + 3z - 4 = 0 Homework Equations N/A The Attempt at a Solution First I'll label them: 2x - 3y - z + 1 = 0 [1] 3x - 2y + 3z - 4 = 0 [2] Then I get rid of the...
  27. mccoy1

    Calculating Volume of Intersection for 3 Balls with Different Centers

    Homework Statement If i have 3 balls of radii =2 and centres =(1,0,0),(0,1,0) and (0,0,1). Find the volume of the intersection of the three balls. Homework Equations The Attempt at a Solution The only method i know only works when the first ball has a centre at (0,0,0) and the...
  28. E

    Evaluating Integrals on a Cone & Plane Intersection

    Homework Statement let c be the curve of intersection of the cone z= sqrt(x^2+y^2) and the plane 3z= y+4, taken once anticlockwise when viewed from above. (i) evaluate ∫c (sinx - y)dx +(x+cosx)dy + (e^z + z)dz (ii) let s be the surface of the cone z= sqrt(x^2+y^2) below the plane 3z=...
  29. L

    Find the intersection of three planes (a line)?

    Homework Statement The following system of equations represents three planes that intersect in a line. 1. 2x+y+z=4 2. x-y+z=p 3. 4x+qy+z=2 Determine p and q 2. The attempt at a solution The problem I have with this question is that you are solving 5 variables with only 3 equations. I...
  30. A

    Intersection points of the planes

    Homework Statement Find all the intersection points of the planes: 2x-y-z=3 x+2y+3z= 7 Homework Equations Whats the best n most simplest way to go about this question. Thanks The Attempt at a Solution
  31. J

    Max Non-Adjacent Vertices from Intersection of Hyperplane & n-Cube

    Hi Taking the intersection of a n-cube with any hyperplane, i would like to know the maximum number X of non adjacent vertices of the cube lying in such intersection. In R2 for instance, i can cut the unit square {(0,0),(1,0),(1,1),(0,1)} with a diagonal line passing through (1,0) and...
  32. A

    Parametric Intersection of Planes P1 and P2

    Calculate in parametric form and describe how the planes intersect Where: P1 = x-3y+5z=6 P2 = 2x-7y+9z=2 My attempt Put planes in matrix form: 1, -3, 5, 6 2, -7, 9, 2 Find Echelon Form 1, -3, 5, 6 0, -1, -1, -10 Z = free variable = a So: -y-z=-10 y = 10 -...
  33. P

    Find intersection of parametric curve and line

    Homework Statement I'm trying to find when that parametric curve intersects with the line x=20 Homework Equations x(t)=(2t^3)/(t^2-1) ; y(t)=(2t^3)/((t^2+1)^2) The Attempt at a Solution I tried representing the line as y=t ; x=20 35=2t^3/(t^2-1) ; t=2t^3/((t^2+1)^2) I also ended up with...
  34. B

    Intersection Form of Connected Sum of CP^2

    Hi, Everyone: Sorry if this is too simple: I guess the intersection for for CP^2 (complex projective 2-space) is (-1), right?. Since H_2(CP^2,Z)=Z, which is represented by CP^1, which has self-intersection=-1. Then, if we had a connected sum of CP^2's, the intersection form...
  35. S

    Fidn intersection of two points parametrically, with two variables

    1. A better way to find the point of intersection of two lines is parametrically as two linear interpolations b/w inital and final points. x=(1-s)x1+sx2 y=(1-s)y1+sy2 where x1 and y1 are the inital points and x2,y2 are the final points. (-6,-6) (5,2) x=(1-t)x3+tx4 y=(1-t)y3+ty4...
  36. H

    Parametric Curve from the intersection of 2 surfaces

    Homework Statement Prove that the curve \vec{r}(t) = <cost,sint/sqrt(2), sint/sqrt(2)> is at the intersection of a sphere and two elliptic cylinders. Reparametrize the curve with respect to arc length measured from (0, 1/sqrt(2), 1/sqrt(2)) in the direction of increasing t. Homework Equations...
  37. M

    Intersection of a Curve and a Surface

    1. At what points does the curve r(t)=ti+2tj+t2k intersect the surface z = x2+y2-100? Give the coordinates of the points. 2. Given equations above. 3. r(t)=<t, 2t, t2> z = x2+y2-100 (t2) = (t)2+(2t)-100 -4t2 = -100 t = sqrt(25) = +/- 5 when t = 5, (5, 10, 25) when t = -5 (-5, -10, 25) This...
  38. L

    Intersection of unindependent events

    Homework Statement Show -0.25 <= P( X \cap Y ) - P( X )P( Y ) <= 0.25 for any events X, Y Homework Equations P( X \cap Y ) = P( X )P( Y | X ) Bayes' theorem Anything I missed? The Attempt at a Solution Obviously if X and Y are independent P( X \cap Y ) = P( X )P( Y )...
  39. B

    Intersection Form With Coeffs. in Z/2

    Hi, Everyone: Just wondering if anyone knew about how to work with the intersection form with coefficients in Z/2. I only know this is in relation to Wu's vector, tho I don't know what Wu's vector is. I was also hoping to know if the intersection form for (4n+2)-manifolds...
  40. B

    Definition of Normal (Intersection) Without Using a Metric

    Hi, Everyone: I am trying to understand the meaning of a statement that two embedded manifolds intersect normally*. The statement is made in a context in which any choice or existence of a metric is not made explicit, nor--from what I can tell-- implicitly either. If...
  41. J

    Point of Intersection of two lines

    Two tangents to an ellipse meet at a point T, find the coordinates of T. The two equations are (bcosΘ)x + (asinΘ)y= ab (-bsinΘ)x + (acosΘ)y= ab This has been really frustrating me as I feel it should be simple, but with the trigonometric...
  42. D

    Intersection of Polynomial and Exponential Functions

    Homework Statement At how many points in the xy-plane do the graphs of y=x^{12} and y=2^{x} intersect? Homework Equations none The Attempt at a Solution I have no idea what to do. I thought of trying to narrow it down to some intervals where the graphs may cross, but, since they're...
  43. mccoy1

    Finding a Non-Trivial Quadratic in the Intersection of Two Subspaces

    Homework Statement I'm given two subspaces L and K of P2 (R) are given by L = { f(x) : 19f(0)+f ' (0) = 0 } K = { f(x) : f(1) = 0 }. Obtain a non-trivial quadratic n = ax2 + b x +c such that n is element of the intersetion of L and K. Homework Equations The...
  44. D

    Dimension of The Intersection of Subspaces

    Homework Statement If V and W are 2-dimensional subspaces of \mathbb{R}^{4}, what are the possible dimensions of the subspace V \cap W? (A). 1 only (B) 2 only (C) 0 and 1 only (D) 0, 1, 2 only (E) 0,1,2,3, and 4 Homework Equations dim(V + W) = dim V + dim W - dim(V \cap W) dim (V + W) \leq...
  45. A

    How do I find the intersection of 4D lines?

    Hello all, Given two 3D lines described by the general equation \vec{L(t)}=\vec{p}+\vec{d}t I found a way to find their intersection point, but it uses the cross product in the derivation. I am assuming a 4D line is a valid thing? And can be described the same way? (except with 4 element...
  46. W

    Intersection of bars moving at v>c

    I have two metal bars positioned in space so that, when viewed in the xy-plane, they intersect each other at some point P. One of the rods are parallel with the x-axis and at rest, while i move the other rod downwards, in the -y direction, with a speed u. The speed of the point P, called U_P...
  47. G

    How Do You Prove Subset Relationships Within Intersecting Indexed Sets?

    Homework Statement Show that the intersection of Ai (for all i in I = {1, 2, 3, ... n } = A1. Ai is a subset of Aj whenever i <= j.Homework Equations The Attempt at a Solution Show: ***I'm having trouble showing part 1***1. that the intersection of Ai is a subset of A1, and 2. A1 is a subset of...
  48. S

    Prove that the intersection of any collection of closed sets in a topological space X

    Prove that the intersection of any collection of closed sets in a topological space X is closed. Homework Statement Homework Equations The Attempt at a Solution
  49. K

    Intersection of axiomatizable sets

    Homework Statement Suppose that T and F are both axiomatizable, complete, consistent theories. Is T\cap F axiomatizable? Homework Equations A theory T is a set of sentences such that if yo can deduce a sentence a from T, then a is in T. I have already proved that T\cap F is a theory...
  50. M

    What is the intersection of nullspaces of S1, S2, and S3?

    I am looking to find a vector which does not lie in various subspaces. For example, if I have: S1 = [1,0,0; 0,1,0] (x-y plane) S2 = [1,0,0; 0,0,1] (x-z plane) S3 = [0,1,0; 0,0,1] (y-z plane) I want to find a vector which was not within any of these subspaces - in this specific example...
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