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. Mutlu CELIKKOL

    Calculating the dimension of intersection of two matrices

    <Moderator's note: Moved from a technical forum and thus no template.> I am at the beginners level of linear algebra and having problem of the intersection of matrices. Your kind help is much appreciated for the following question Let\quad M1=\begin{Bmatrix} x & -x \\ y & z \end{Bmatrix},\quad...
  2. Mr Davis 97

    Intersection of any 2 intervals --> all intervals intersect

    Homework Statement Let ##\mathscr{F}## be a finite family of open or closed intervals in the line ##\mathbb{R}^1##. Show by an elementary proof that if any ##2## of them intersect, then all of them intersect. Homework EquationsThe Attempt at a Solution Here is my attempt, where for now I'm...
  3. Loubrainz

    How do I find the intersection of three cones?

    Homework Statement tl;dr: looking for a way to find the intersection of three cones. I'm currently working with a team to build a Compton camera and I've taken up the deadly task of image reconstruction. Background Theory: https://en.wikipedia.org/wiki/Compton_scattering For a single Compton...
  4. Krushnaraj Pandya

    Line of intersection of two planes

    Homework Statement Find equation of line of intersection of planes r.(3i-j+k)=1 and r.((i+4j-2k)=2 Homework Equations a x b gives a perpendicular vector to a and b...(i) The Attempt at a Solution to write equation of a line r=a+tb we need a point a on the line and a parallel vector b. Taking...
  5. Mr Davis 97

    Using class equation to show that intersection is nontrivial

    Homework Statement Let ##G## be a finite p-group and ##Z(G)## its center. If ##N \not = \{e\}## is a normal subgroup of ##G##, prove that ##N\cap Z(G) \not = \{e\}##.Homework EquationsThe Attempt at a Solution Since ##N## is a normal subgroup we can let ##G## act on ##N## by conjugation. In a...
  6. M

    MHB Intersection points of two quadratic functions

    Justify the following by using table, graph and equation. use words to explain each representation f(X) = 2 x2 - 8x and g(x) = x2-3x+ 6 the points (-1,10) and (6,24)
  7. Z

    Kinematics Problem: Train braking and blocking an intersection

    Homework Statement A train 400 m long is moving on a straight track with a speed of 21.6m/s. The engineer applies the brakes at a crossing, and later the last car passes the crossing with a speed of 5.0m/s. Assuming constant acceleration, determine how long the train blocked the crossing...
  8. Adgorn

    Spivak: Conic Sections appendix, problem 1

    Homework Statement "Consider a cylinder with a generator perpendicular to the horizontal plane; the only requirement for a point ##(x,y,z)## to lie on this cylinder is that ##(x,y## lies on a circle: ##x^2+y^2=C^2##. Show that the intersection of a plane with this cylinder can be described by...
  9. L

    MHB Intersection between line and plane in R4-RN

    Hello, I believe that this is the correct forum for this post but please let me know if otherwise. I have the following data in R4, id x y z a B -0.0700177 0.382249 -0.289338 0.00349957 A 1.13388 1.77602 -0.679365...
  10. bornofflame

    [Linear Algebra] Show that H ∩ K is a subspace of V

    Homework Statement From Linear Algebra and Its Applications, 5th Edition, David Lay Chapter 4, Section 1, Question 32 Let H and K be subspaces of a vector space V. The intersection of H and K is the set of v in V that belong to both H and K. Show that H ∩ K is a subspace of V. (See figure.)...
  11. T

    MHB Finding Coordinates of Intersection in Parametric Forms

    Hey, I have a couple of questions I've been doing online which have left me a little puzzled. The first one, I'm not really sure how to go about. I think a lot of that comes down to having not had a lot of experience with parametric forms. I'll just post screenshots of where I'm up to on...
  12. Physics345

    Intersection of two lines in 3-space

    Homework Statement Given l→1=(6,−1,0)+t(3,1,−4) and l→2=(4,0,5)+s(−1,1,5), find the intersection of l→1 and l→2 . Homework EquationsThe Attempt at a Solution I'm fairly certain I did this correctly, but I just thought I'd double check to make sure I have a good understanding.
  13. O

    MHB Finding Points of Intersection for Trigonometric Functions

    does anybody know how to solve this problem ?Consider two trigonometric functions y = cos 2x and y = 1 + sinx.(a) Write down the equation in x that you would solve to find the x coordinate of point(s) of intersection of those graphs on [0, 2∏]. (b) Solve your equation, and write down the...
  14. SuchBants

    Uncertainty in the intersection of two lines with error bars

    The only thing I wish to take from this graph is the intersect of the two lines of best fit. How can I get the uncertainty in this point? While the points appear to have the same uncertainties, they actually each have their own distinct % uncertainty. Would the maximum uncertainty in the X...
  15. M

    Find the point of intersection?

    Homework Statement Find the point of intersection of the following lines, if it exists. Homework Equations None. The Attempt at a Solution I think the first step is to form 2 equations, but I'm not too sure.
  16. heavystray

    B How do you calculate the uncertainty in T using graphs?

    i have an equation like this: Given m(B-V)= 1.2 +- 0.2 How do you calculate the uncertainty in T? (btw, I solve T using graphs by finding the intersection point) My idea was first to calculate T when m(B-V) =0.2, and I then calculate T when m(B-V)= 1.2 + 0.2(its uncertainty). and then find the...
  17. F

    Find intersections between sphere and parallel tangent planes

    Mod note: Moved from a technical forum section, so missing the homework template. @fab13 -- please post homework problems in the appropriate section under Homework & Coursework. I have the following exercise to solve : I have to find all the points on the surface ##x^2+y^2+z^2=36## (so a sphere...
  18. O

    Finding the Line of Intersection for Two Cylinders

    Homework Statement I am looking for the line of intersection of two cylinders (x^2+y^=1 and x^2+z^2=1) to find the curve for a line integral. However, I am getting z=y, which is a patently false answer. Mathematically, it seems to work out though. Homework Equations x^2+y^=1 x^2+z^2=1 The...
  19. M

    MHB Intersection of all subspace of V is the empty set

    Hey! :o Let $V$ be a $\mathbb{R}$-subspace with basis $B=\{v_1 ,v_2, \ldots , v_n\}$ and $\overline{v}\in V$, $\overline{v}\neq 0$. I have shown that if we exchange $\overline{v}$ with an element $v_i\in B$ we get again a basis. How can we show, using this fact, that the intersection of all...
  20. L

    MHB Formula for changing a plane intersection angle

    Hello, This is a bit more complex than my previous post, but I don't think it qualifies for university level difficulty. Please move this to the appropriate forum if my assessment is not correct. I have 4 vectors in three space where each vector has its tail at the same point (point B). id...
  21. B

    Intersection of sets in ##\Bbb{R}^2## and Open Maps

    Homework Statement No problem statement Homework EquationsThe Attempt at a Solution Let ##A = \{(x,y) ~|~ x \ge 0 \mbox{ or } y=0 \}## be a subspace, which can be shown closed, of ##\Bbb{R}^2##. If my calculations are right, isn't ##A \cap [(0,\infty) \times \Bbb{R} ] = (\{0\} \times...
  22. peroAlex

    Parametrize the Curve of Intersection

    Hi everyone! I'm a student of electrical engineering. At my math class, we were given a problem to solve at home. Now, from what I've managed to gather, this is a trick question, but I would like to get someone else's opinion on the task. It's also worth mentioning that parametrization is a...
  23. E

    Intro statistics question: probability of intersection

    Homework Statement If event A equals event B, then the probability of their intersection is 1. True or False? Apparently the correct answer is False. The Attempt at a Solution If A=B then they should overlap entirely and their intersection should be 1? The only way I see this working is if...
  24. B

    Conditions for Strict Inequality in Span Intersection

    Homework Statement Let ##S_1## and ##S_2## be subsets of the vector space ##V##. Prove that ##span (S_1 \cap S_2) \subseteq span(S_1) \cap span(S_2)##. Given an examples of ##S_1## and ##S_2## for which equality holds and for which the inequality is strict. Homework EquationsThe Attempt at a...
  25. karush

    MHB 243.11.5.9 Area of intersection cardioid and circle

    OK just seeing if this is setup OK before I pursue all the steps I thot adding areas would be easier:cool:
  26. Joacim Jacobsen

    I Intersection between line and cylinder

    I have an expression for a "line- to sphere intersection" that works: a = 1 + Ax^2 + Ay^2 b = 2*(-zs + Ax*(Bx-xs) + Ay*(By-ys)) c = zs^2 + (Bx-xs)^2 + (By - ys)^2 - R^2 This is part of a code in Matlab, and works fine. It is derived from substituting (x=Ax*z+Bx, y=Ay*z+By) into...
  27. R

    Exploring the Intersection of Physics & Math: Are We Going More Mathematical?

    In physics are we going more mathematical?
  28. 0

    Intersection of Lines: Solving for m and n to Determine Concurrent Lines

    Homework Statement For which m and n the lines are concurrent? ##r: \begin{cases} x & - & y & = & 1 \\ nx & - & y & - & 2z & + & m & + & 1 & = & 0\end{cases}## ##s: \begin{cases} x & - & nz & + & m & + & n & = & 0 \\ x & + & y & - & 2nz & + & 11 & = & 0 \end{cases}## Solving r gives me...
  29. Eclair_de_XII

    Proving that the intersection of two sets is a subset of Q?

    Homework Statement "Let ##U={p+r\sqrt{2}:p,r∈ℚ}## and let ##V={a+b\sqrt{7}:a,b∈ℚ}##. Show that ##ℚ⊆U∩V##. Then show that ##U∩V⊆ℚ## and conclude that ##U∩V=ℚ##." Homework EquationsThe Attempt at a Solution I only knew how to prove the first part. That is: (1) "Suppose that ##ℚ⊄U∩V##. This...
  30. Mr Davis 97

    Tangent vector on the intersection of surfaces

    Homework Statement The surfaces ##x^2+y^2 = 2## and ##y=z## intersect in a curve ##C##. Find a unit tangent vector to the curve ##C## at the point ##(1,1,1)##. Homework EquationsThe Attempt at a Solution So I'm thinking that we can parametrize the surfaces to get a vector for the curve ##C##...
  31. B

    I Ave length of intersection of all lines through a unit cube

    I'm having trouble even beginning to figure out how to approach solutions for this. I begin with a unit cube, and imagine all the possible lines that intersect the cube. I am assuming there must be an average length of these intersections; I want to find that average length. Another way to...
  32. Zero2Infinity

    Basis of the intersection of two spaces

    Homework Statement Consider two vector spaces ##A=span\{(1,1,0),(0,2,0)\}## and ##B=\{(x,y,z)\in\mathbb{R}^3 s.t. x-y=0\}##. Find a basis of ##A\cap B##. I get the solution but I also inferred it without all the calculations. Is my reasoning correct Homework Equations linear dependence...
  33. B

    Index of Intersection of Subgroups with Finite Index

    Homework Statement Suppose that ##H## and ##K## are subgroups of finite index in the (possibly infinite) group ##G## with ##|G : H|m## and ##|G:K|=n##. Prove that ##lcm(m,n) \le |G : H \cap K | < mn##. Homework EquationsThe Attempt at a Solution I was able to get the upper bound on ##|G : H...
  34. F

    MHB Intersection of Sets A, B and C in ℤ

    Let A,B,C be three sets such that : A={x∈ ℤ / x=11k+8 , k∈ℤ} B={x∈ ℤ / x=4k , k∈ℤ} C={x∈ ℤ / x=11(4k+1) -3 , k∈ℤ } Prove A⋂B = C I started with this : Let x be an arbitrary element of A⋂B then ∃(k,k')∈ ℤ² such that x=11k+8 and x=4k' then 11k+8 = 4k' then 11(k+1)-3 = 4k' I don't know where...
  35. Kernul

    Checking if intersection is correct

    Homework Statement The exercise gives me ##V = <(1, 0, 0, 1), (-7, 0, 1, 0)>## and ##U = <(0, 4, 1, -1), (1, 12, 0, 26)>## and I have to find dimension and a base of ##V \cap U## and ##V + U##. Homework EquationsThe Attempt at a Solution They both have dimension ##2##. I find a vector ##\vec v...
  36. Mr Davis 97

    Describing the intersection of two subspaces

    Homework Statement ##W_1 = \{(a_1, a_2, a_3) \in \mathbb{R}^3 : a_1 = 3a_3,~ a_3 = -a_2 \}## ##W_2 = \{(a_1, a_2, a_3) \in \mathbb{R}^3 : 2a_1 - 7a_2 + a_3 = 0 \}## Given that these are two subspaces of ##\mathbb{R}^3##, describe the intersection of the two, i.e. ##W_1 \cap W_2## and show that...
  37. Bishamonten

    Finding the point on the line of intersection between planes

    Homework Statement Given two planes, P1, P2. Find parametric equations for the line of intersection between the two planes. Homework Equations P1 = 2x -3y + 4z = 3 P2 = x + 4y - 2z = 7 The Attempt at a Solution Let N1 be the normal vector to P1, and N2 be the normal vector to P2. Then, N1...
  38. Saracen Rue

    B Intersecting Bivariate Functions: Is x=a the Only Point of Intersection?

    Will the bivariate function ##f(x,a)## always intersect ##f(a,x)## at the point ##x=a## given that ##f## is a real, defined function? (other points of intersection can exist but are not relevant for this question)
  39. Kernul

    Intersection of Subspaces in R^4: Finding Bases and Dimensions

    Homework Statement I have this exercise that tells me to determine a base and the dimension of the subspaces of ##\mathbb {R}^4##, ##U \cap Ker(f)## and ##U + Ker(f)##, knowing that: ##U = <\begin{pmatrix} -10 \\ 11 \\ 2 \\ 9 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \\ 1 \\ 3 \end{pmatrix}...
  40. arupel

    I Intersection & union of closed and open sets?

    I am a little confused here: a) The number 2 which is at the beginng of one set is closed. The number 2 is open at the beginning of the other set. b) The number 2 is closed of the beginning of a set which goes to infinity. The other set begins at 0 and goes to infinity (2 is an...
  41. Erenjaeger

    I The angle of intersection between two planes in R3

    What is the angle of intersection between the two planes in ℝ3 with general equations x-y=5 and y-z=7? I know that the angle between then is equal to cos-1 (u⋅v/||u|| ||v||) but I am stuck on the general equations given, how can I solve when given these ?
  42. F

    MHB Re: Union and Intersection of Sets

    Re: Union and Intersection of Sets Hi, Please I need a help regarding Union of sets can anybody solve this A={1,2,3} and B={{1,2},3} then what is A Union B and A Intersect B Thanks
  43. EternusVia

    Programs Intersection of Physics and Applied Math?

    Hello all, I am interested in both physics and applied math. One of my professors is willing to work with me to get into a great applied math school, but I don't want to abandon my passion for physics. Is there work being done at the intersection of applied math and physics? Is it reasonable...
  44. mr.tea

    I Cantor's intersection theorem (Apostol)

    Hi, I am reading "mathematical analysis" by Apostol right now for a course in analysis. Since I am trying to understand the author's proof of the above theorem(3.25 in the book), but I have something that I can't understand. He assumes that each of the nested sets contains infinitely many...
  45. M

    MHB Show that the intersection is a pp -Sylow subgroup

    Hey! :o I want to show that if $S\in \text{Syl}_p(G)$ and $N\trianglelefteq G$, then $N\cap S\in \text{Syl}_p(N)$. Could you give me some hints how we could show that? (Wondering) Do we maybe use Frattini's Argument? (Wondering) From that we have that since $N\trianglelefteq G$ and $S\in...
  46. Kernul

    Exercise with intersection and sum

    My professor did this exercise that I didn't quite get how she went through all of it. We have a ##U = {(x, y, z, t) : x+y+z+t = 0}## and ##B_{Im(f)} = \left[ \begin{pmatrix} 7 \\ -3 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 3 \\ -3 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 5 \\ 0 \\ 1 \\ -5...
  47. evinda

    MHB Where Do the Diagonals of a Parallelogram Intersect?

    Hello! (Wave) Suppose that a parallelogram $ABCD$ has vertices $A=(0,0)$ and $B=(1,0)$. In terms of $C=(x,y)$, find the position of $D$ and where the two diagonals will intersect. Then we will have something like that: How can we find the coordinates of $M$, i.e. the point at which the two...
  48. G

    Question about equations of 3D solids

    I was working on a problem on this domain: E=[x,y,z)\:s.t. \: \sqrt{x^2+y^2}\leq z\leq \sqrt{3x^2+3y^2},\: x^2+y^2+z^2\leq 2] and at some point I wanted to find the intersection of the internal cone(##\sqrt{3x^2+3y^2}=z##) with the sphere of radius ##\sqrt{2}## to find the height z of the...
  49. S

    MHB Line of intersection between two parallel planes?

    How can I find the line of intersection between the planes 2x-y+2x+1=0 and -4x+2y-4x-2=0 I realize these are parallel as they are multiples of each other, but I'm not sure how to solve for the point. I also have to convert this line into parametric, cartesian and vector form. Sorry for the...
  50. 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...
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