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

    MHB Solving the nature of intersection between 3 planes

    Stuck on the last 3 systems on my worksheet, if someone could give me more than just the answer but also teach me how, it would be much appreciated! "Determine the nature of the intersection if it exists) between the following sets of planes. If it is a line, find the equation of it. If it is a...
  2. S

    Issue with behavior of ray-plane intersection algorithm

    I'm trying to determine the point, in 3D space, where an arbitrary line/ray intersects with an infinite plane. Using an article on Wikipedia, I tried to reproduce the presented formulas in code. This seems to work fine so long the ray is emitted from the origin of the coordinate system (0, 0...
  3. M

    Quadratic and linear Intersection

    Homework Statement How many points of intersection does the line y = x have with the parabola y = 6x − x2 −10 ? A) none B) one C) two D) three E) more than three Homework EquationsThe Attempt at a Solution x = 6x - x^2 - 10 -x^2+5x - 10 Used b^2 - 4ac to check the number of intersections. The...
  4. Rumplestiltskin

    Integrating for area of intersection

    How does it work that you can subtract y2 from y1 and integrate the product within defined limits for the area of their intersection (within those limits)? Maybe that's not the right terminology - you arrive at the area for the region bounded by both functions. Is it just the same in practice...
  5. S

    Intersection of general plane with quadric surface

    So far I have calculated this for a cone, which elegantly results in the conic sections. However, I would like to do this for the other quadric surfaces. Is the calculation for this been published anywhere online?
  6. G

    Local Algebra: intersection multiplicity

    I'm trying to understand why the intersection multiplicity of two singular subvarieties is not equal to the complex dimension of the local ring but it is instead the Euler characteristic. Is it possible to find an intuitive explanation? I think that the following concepts need some...
  7. A

    Intersection of Hyperboloid & 2-Plane=Ellipse

    I want to try and see the intersection between the hyperboloid and the 2-plane giving an ellipse. So far I have the following: I'm going to work with ##AdS_3## for simplicity which is the hyperboloid given by the surface (see eqn 10 in above notes for reason) ##X_0^2-X_1^2-X_2^2+X_3^2=L^2## If...
  8. S

    Finding the Intersection of a Sinusoidal Function and a Line

    Homework Statement Hi! I'm trying to find the points of intersection of a sinusoidal function and a line. The line is y=x/7. The function is y=sinx. Can someone tell me how to determine the number of intersections and exact intersections. I would also like to know if the same method can be...
  9. Conservation

    Finding the intersection of an ellipsoid and a plane

    Homework Statement Find the curve that is the intersection of x-y-z>-10 and x2+y2/4+z2/9=36. Homework EquationsThe Attempt at a Solution The best idea I have is to define x as x=y+z-10 and substitute it into the ellipsoid equation to get a function defined by y and z; the trouble is that...
  10. K

    Intersection Line for Two Planes

    Homework Statement Find an equation of the line where the planes Q and R intersect. Q: -2x + 3y - z = 1; R: x + y + z = 0 Homework Equations Equation of a Plane: ax + by + cz = d, where \vec{n} = <a, b, c> Equation of a Line in R^{3}: \vec{r}(t)=<x_{0}, y_{0}, z_{0}> + t<x,y,z> The Attempt at...
  11. goonking

    Intersection of line through a plane

    Homework Statement Where does the line through (−2, 3, 2) and (3, 5, −1) intersect the plane x + y − 2z = 6? Homework EquationsThe Attempt at a Solution i used r = r0 + tv the vector between the 2 given points is <5,2,-3> r = (-2,3,2) + t<5,2-3> x = -2 + 5t y=3+2t z=2 - 3t plugging these...
  12. C

    MHB Points of intersection of two vector curves

    a) At what point do the curves r1(t) = (t, 2 − t, 35 + t2) and r2(s) = (7 − s, s − 5, s2) intersect? (x,y,z) = b) Find their angle of intersection, θ, correct to the nearest degree.
  13. Titan97

    Question on intersection of tangent and chord

    Homework Statement Show that The tangent at (c,ec) on the curve y=ex intersects the chord joining the points (c-1,ec-1) and (c+1,ec+1) at the left of x=c Homework Equations Legrange's mean value theorem The Attempt at a Solution f'(c)=ec Applying LMVT at c-1, c+1...
  14. S

    How do I parameterize the intersection of these two surfaces?

    Homework Statement Parameterize the curve of intersection of the two surfaces: x^2+y^2+z^2=14 z=y^2-x^2 Homework EquationsThe Attempt at a Solution I tried manipulating the equations above but can't seem to get a nice parameterization which I can use to do the rest of the (calculus) problem.
  15. Andrew Pierce

    How to find a point of intersection of two planes?

    Homework Statement Two planes are given by the equations x + y + z = 1 for the plane P1 and x − y + z = 1 for the plane P2. (or) P1 : x + y + z = 1 P2 : x - y + z = 1 Q. The Question Find the coordinates of a point of intersection of the planes P1 and P2 Homework...
  16. W

    Transversal Intersection of More than 2 Surfaces

    Hi, There is a result that if two manifolds ## M_1, M_2 ## ( I don't know to what extent this generalizes to other topological spaces) intersect transversally, say in ##\mathbb R^m ## , then the dimension of the intersecting set is given by m - ##\Sigma Cod(M_i ) ; i=1,2##, where ##Cod(M_i):=...
  17. P

    Point of Intersection involving logarithms

    Homework Statement Solve for the point of intersection between ##y=\log_{2}{(2x)}## and ##y=\log_{4}{(x)}## Homework Equations 3. The Attempt at a Solution [/B] Setting the two equations equal: $$\log_{2}{(2x)}=\log_{4}{(x)}\\ 2x=2^{\log_{4}{(x)}} \\ 2^{2x}=4^{\log_{4}{(x)}}\\...
  18. J

    Mathematica Mathematica: supply list of list to Intersection?

    I notice there are some functions in Mathematica in which you can't supply directly, an array of arguments but rather must give it a list. For example, GCD and LCM. I can use for example, GCD[2,3,8] and it returns 1. However if I code: GCD[{2,3,8}], the function call fails. I know of one way...
  19. S

    DeMorgan's Law extended to Union AND Intersection

    Hello. We all know that DeMorgan's Law is as follows: (A∪B)' = A'∩B' and (A∩B)' = A'∪B' where ' refers to the complement of a set and A and B are both sets. We also know that this can be extended to more than two terms. My question is whether or not the following is true: (A∩B∪C)' = A'∪B'∩C'...
  20. M

    Intersection of line and surface

    A straight line in 3 space can be described as A + Bt, where A is a position, B a direction, and t a scalar parameter. CAD surfaces can be represented in terms of polynomial functions of two variables (u and v) with the highest degree term being u^nv^n. The intersections can then be obtained as...
  21. C

    Intersection of two parabolas where one is vertex shifted

    Not a homework question but this seems to be the appropriate place... 1. Homework Statement I would like to be able to calculate the intersections of two parabola's which accounts for one or both of the parabola's being shifted along the x axis I have written an excel vba function to do this...
  22. B

    Set theory, intersection of two sets

    Homework Statement We have the set D which consists of x, where x is a prime number. We also have the set F, which consists of x, belongs to the natural numbers (positive numbers 1, 2, 3, 4, 5..) that is congruent with 1 (modulo 8). What numbers are in the intersection of these two sets...
  23. F

    Orthogonal complement of the intersection of 2 planes

    Homework Statement Let W be the intersection of the two planes: x-y+z=0 and x+y+z=0 Find a basis for and the dimension of the orthogonal complement, W⊥ Homework EquationsThe Attempt at a Solution The line x+z=0 intersects the plane, which is parameterized as t(1, 0, -1) Then W⊥ is the plane...
  24. B

    MHB Plane that is tangent to two curves at an intersection

    Can someone please help me with how to approach/solve this question? construct a plane that is tangent to both curves at the point of intersection. 1st curve: x(v)=3 y(v)=4 z(v)=v 0<v<2 2nd curve: x(u)=3+sin(u) y(u)=4−u z(u)=1−u −1<u<1 My first approach was to find a point of intersection...
  25. M

    Finding Points of Intersection for Two Functions

    Homework Statement The problem ask for points of intersection of two functions Homework Equations 1: 2x+y-4=0 2: (y^2)-4x=0 The Attempt at a Solution My attempt of solution its in a picture attached below... I get stuck in this two equations 1: ((y^2)/4)+(y/2)-2=0 2: square...
  26. Alexander1

    MHB Finding the intersection points on the graph y=sinx, y=cosx and y=tanx

    Hi guys, I'm new to this site and it seems like it will be a great resource when I'm stuck on a problem. I'll firstly set out the question and then add in my working so far. Question: I was firstly asked to graph the trigonometric functions y=sinx, y=cosx and y=tanx in the interval where x is...
  27. N

    How to find a point on line of intersection of 2 planes?

    Hey all, for some software I'm writing a sub problem of a bigger math problem I have is that I need to find the line of intersection of two planes, One can obtain the normal via the cross product but I am stuck at how to find a point on that line as they're seems to be too many variables...
  28. B

    Parametrization of a curve(the intersection of two surfaces)

    Homework Statement I am looking to find the parametrization of the curve found by the intersection of two surfaces. The surfaces are defined by the following equations: z=x^2-y^2 and z=x^2+xy-1 Homework EquationsThe Attempt at a Solution I can't seem to separate the variables well...
  29. C

    How Does One Deduce Area of Intersection for Three Cirlces?

    Homework Statement Homework Equations Most likely Acircle = πr2 Not sure if there's others. The Attempt at a Solution I'm not sure where to start, I've never seen a question of this sort. They all have the same radius, hence same area, and each point/center is r away from another, but I don't...
  30. M

    MHB Level Surfaces & Intersection of a Graph: Exploring $f(x,y,z) = x^2+y^2$

    Hey! :o Draw or describe the level surface and an intersection of the graph for the function $$f: \mathbb{R}^3 \rightarrow \mathbb{R}, (x, y, z) \rightarrow x^2+y^2$$ I have done the following: The level surfaces are defined by $$\{(x, y, z) \mid x^2+y^2=c\}$$ - For $c=0$ we have that...
  31. M

    MHB Find the Intersection Point of Lines $(3,1,-2)$ and $x=-1+t, y=-2+t, z=-1+t$

    Hello! :o I have to find the line that passes through $(3, 1, -2)$ and intersects under right angle the line $x=-1+t, y=-2+t, z=-1+t$. (HINT: If $(x_0, y_0, z_0)$ the intersection point, find the coordinates.) I have done the following: The line that passes through $(3, 1, -2)$ is of the...
  32. RJLiberator

    How to identify the curve at intersection of level surfaces

    Homework Statement Sketch a picture of the cone x = sqrt(y^2+z^2) , and elliptic paraboloid x = 2−y^2−z^2 on the same grid. Although the picture does not have to be perfect, indicate clearly the orientation of both figures relative to coordinate axes. Identify the curve at the intersection of...
  33. C

    Intersection between exponential models?

    Homework Statement Here is a math problem I would to make sure I have understood correctly. a) Billy goes to be bank. He deposite 500 dollars in his account. The bank offers a interest of 2 percent a year. Sally goes to the bank 4 years later.She deposite 250 dollars at the same bank...
  34. Calpalned

    Intersection of a circle with coordinate planes

    Homework Statement Find an equation of the sphere with center (2, -6, 4) and radius 5. Describe its intersection with the each of the coordinate planes. Homework Equations Equation of a sphere with three dimensions X2 + Y2 + Z2 = R2 The Attempt at a Solution My equation is (x - 2)2 + (y +...
  35. nuuskur

    Difference unexpressable as intersection

    Is there anywhere to look for proofs that deal in tackling the set theoretic operations - what can and what can't be expressed through another operation? For example: prove that the difference of 2 (or N) sets cannot be expressed through the intersection operator. Given A,B we cannot I'm not...
  36. M

    Curve of intersection of a plane and function

    Homework Statement f(x,y) = x^2 + xy + y^2 and z=2x+y intersect, find a parameterization of the curve where they intersect. Homework EquationsThe Attempt at a Solution I am lost. I know that z is the partial derivative of the original function, if that's of any use. I can visualize it but not...
  37. S

    Sample spaces, events and set theory intersection

    Homework Statement Problem: Given a regular deck of 52 cards, let A be the event {king is drawn} or simply {king} and B the event {club is drawn} or simply {club}. Describe the event A ∪ B Solution: A ∪ B = {either king or club or both (where "both" means "king of clubs")} Homework Equations...
  38. F

    Intersection of a closed convex set

    Let X be a real Banach Space, C be a closed convex subset of X. Define Lc = {f: f - a ∈ X* for some real number a and f(x) ≥ 0 for all x ∈ C} (X* is the dual space of X) Using a version of the Hahn - Banach Theorem to show that C = ∩ {x ∈ X: f(x) ≥ 0} with the index f ∈ Lc under the...
  39. Bassa

    The Line of Intersection of Two Planes

    Homework Statement Find a set of parametric equations for the line of intersection of the planes. 6x-3y+z=5 and -x+y+5z=5[/B]Homework Equations The cross product formula The formula for the parametric equations of a line in three dimensional space: x=x1+at, y=y1+bt, z=z1+ct Knowing the fact...
  40. Matterwave

    Are Jacobi fields defined at intersection points?

    I have some questions with regards to conjugate points on a congruence of time-like geodesics (will be referring to Wald 9.3 throughout). First, we define ##\gamma## to be a time-like geodesic with tangent ##\xi^a## parametrized by ##\tau## and with ##p\in\gamma##. We consider the "congruence of...
  41. M

    MHB Finding the Intersection of Subfields in Finite Fields

    Hey! :o Let $\mathbb{F}_{p^m}$ and $\mathbb{F}_{p^n}$ be subfields of $\overline{\mathbb{Z}}_p$ with $p^n$ and $p^m$ elements respectively. To find the field $\mathbb{F}_{p^m} \cap \mathbb{F}_{p^n}$ I have done the following: Let $\mathbb{F}_{p^n} \cap \mathbb{F}_{p^m} = \mathbb{F}_{p^d}$...
  42. D

    Questions about the intersection between chemistry & physics

    I am in some real need of advice on the possibilities of my future plans. A bit of background, 32 years old, went back to undergrad at 30 and currently enrolled in a Bsc (hons) degree in Natural Sciences in the UK with my major subject as Chemistry. Plans to give my full effort to getting into...
  43. evinda

    MHB Union or Intersection for f(x)=0 When x in A and B

    Hello! (Wave) When we have: $f(x)=0, \forall x \in A \wedge f(x)=0, \forall x \in B$, do we conclude that $f(x)=0, \forall x \in A \cap B$ or $f(x)=0, \forall x \in A \cup B$? (Thinking)
  44. evinda

    MHB Proving Intersection of Ideals in $K[x_1,x_2,...,x_n]$

    Hi! (Smile) Let $(I_a)_{a \in A}$ be a family of ideals of $K[x_1,x_2, \dots, x_n]$. I want to prove that: $$V \left ( \sum_{a \in A} I_a\right )=\bigcap_{a \in A} V(I_a)$$ Do we have to use the definition: $$V(S)=\{ (a_1,a_2, \dots, a_n) \in K^n| f_a(a_1,a_2, \dots, a_n)=0 \forall a \in...
  45. E

    MHB Distance Between Intersection Points of Bisectors & Medians in Right Triangle

    The legs of chateti of a right triangle are 9 and 12 cm. Find the distance between the intersection point of bisectors and the point of intersection of the medians
  46. P

    Intersection of A & B: Answer to Puzzling Question

    Homework Statement Set A has twice the number of elements as Set B, 1/3 of the elements of Set A are the same as in Set B, the union of A and B is 42, what is the intersection? The Attempt at a Solution This was one of my exam questions, and I just want to see what the correct answer was...
  47. S

    Examples of A intersection B being an element of A

    What are some examples where the intersection of two sets is a member of one of the sets? Let A,B,C,D be sets whose elements are sets of integers. A = \{\emptyset, \{1,2\},\{3\} \} B = \{\{4,5,6\},\{7,8\} \} C = \{ \emptyset, \{7,8\} \} D = \{ \{3\}, \{4,5,6\} \} Then A \cap B =...
  48. Logan Land

    MHB Find basis and dimension of the intersection of S and T

    If you have a vector space S be spanned by vectors (x1,y1,z1), (x2, y2, z2), (x3, y3, z3) and T spanned by (x1,y1,z1), (x2, y2, z2), (x3, y3, z3). How would you find the basis and dimension of the intersection of S and T . (x,y,z can be any value) Do I go about it like this? a(x1,y1,z1)+b(x2...
  49. M

    Line of intersection of two planes

    Hi, I was doing a L.A question and a question arose. ( well I will write the question now, I found the answer I just can't visualize what I am doing which bothers me greatly) Find the equation of the plane that contains the line (x,y,z)=(1,0,0)+t(1,3,2), and is parallel to the line of...
  50. I

    Graphing Intersection Points for y=3x^2 and y=3^x | Algebra Homework Help

    Homework Statement Find the intersection points of y=3x^2 and y=3^xHomework Equations They must be found with graphing techniques and cannot be proved algebraically. The answers are (-.451,.0609), (1.3), and (3, 27). The Attempt at a Solution Table of values from -3 to 3 for x, and i can...
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