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. Yae Miteo

    What is the vector function for the intersection of a cone and a plane?

    Homework Statement "Find a vector function that represents the curve of intersection of the two surfaces." Homework Equations Cone: z = \sqrt{x^2 + y^2} Plane: z = 1+y The Attempt at a Solution I began by setting x=cos t, so that y = sin t and z = 1+sin t. At this point...
  2. P

    Intersection between rotated & translated ellipse and line

    I have a rotated ellipse, not centered at the origin, defined by x,y,a,b and angle. Then I have a segment defined by two points x1,y1 and x2,y2 Is there a quick way to find the intersection points? I used wolfram alpha equation solver, I tried to insert the equation of a line into the one of a...
  3. D

    Intersection between hyperplane within a simplex

    Hi, If we contract a (n-1)d hyperplane with a n-simplex, then what is maximum number of intersection points with the egdes of the simplex and the hyperplane ? For, if we draw a line within a 2-simplex (there are 3 edges), it will have a intersection of maximum two edges. For 3-simplex, any...
  4. Dethrone

    MHB Points of intersection of polar equations

    Find the points of intersection of $\rho=\cos\left({2\theta}\right)$ and $\rho=\cos\left({\theta}\right)$ By setting $\cos\left({2\theta}\right)=\cos\left({\theta}\right)$, we get the solutions $\theta=0,\frac{2\pi}{3},\frac{4\pi}{3}$. My question is how come that doesn't give us all the...
  5. D

    Polar Coordinates, intersection of a cylinder with a spher

    Homework Statement Find the Volume of the solid that the cylinder ##r = acos\theta## cuts out of the sphere of radius a centered at the origin.Homework Equations The Attempt at a Solution I have defined the polar region as follows, $$D = \{ (r,\theta) | -\pi/2 ≤ \theta ≤ \pi/2 , 0 ≤ r...
  6. P

    How to Calculate the Intersection Point of Two Cycloids?

    Homework Statement Greetings everyone! I would like to know, how to calculate the intersection point of two cycloids. Homework Equations The equations of the cycloid are the following: x=r(t-sin(t)) y=r(1-cos(t)) The Attempt at a Solution I tried to solve it by myself, but I...
  7. Greg Bernhardt

    Understanding Conic Sections: Exploring the Plane-Cone Intersection

    Definition/Summary A conic section (or conic) is any curve which results from a plane slicing through an upright circular cone. If the slope of the plane is zero, it cuts only one half of the cone, and the conic is a circle (or a point, if the plane goes through the apex of the cone). If the...
  8. K

    Finding points of intersection algebraically between 2 trig functions

    So I have several problems that ask me to find all points of intersection algebraically, but I haven't been able to make much headway on most of them. The first problem Homework Statement Find all the points of intersection algebraically of the graphs of ... on the interval [0, 4π]...
  9. FysixFox

    Finding d in Circle-Circle Intersection Equation

    Okay, so I've found out about how circle-circle intersection works ( http://mathworld.wolfram.com/Circle-CircleIntersection.html ). I'm working with the following knowledge: The area of the overlap is 100 The two circles have the same radius, 12 d is unknown How would I solve for d in the...
  10. C

    Intersection of an ideal with a subring (B)

    In a previous thread, I asked a question different from that I actually intended to ask. Since this question is licit and was answered by micromass, I open this new thread. The right question is in fact: If R is an integral domain, and R' is INTEGRAL over R, then the function f which assigns...
  11. C

    Intersection of ideals with subring

    Hello, Thanks to the help of micromass in a previous thread, I am now able to prove the following theorem (which can be seen as a (somewhat improved) version of the "going up" and "going down" theorems): If R is an integral domain, and R' is integrally closed over R, then the function f...
  12. Serious Max

    Problem involving intersection of a line and a circle

    Homework Statement Erik’s disabled sailboat is floating stationary 3 miles East and 2 miles North of Kingston. The sailboat has a radar scope that will detect any object within 3 miles of the sailboat. A ferry leaves Kingston heading toward Edmonds at 12 mph. Edmonds is 6 miles due east of...
  13. C

    Intersection of maximal ideal with subring

    Let R be an integral ring (eventually can be supposed integrally closed), and R' an integral extension of R. Assume that M is a maximal ideal of R'. Must the intersection of M with R be a maximal ideal of R ? Thx.
  14. M

    Dual basis and kernel intersection

    The problem statement, all variable Let ##\phi_1,...,\phi_n \in V^*## all different from the zero functional. Prove that ##\{\phi_1,...,\phi_n\}## is basis of ##V^*## if and only if ##\bigcap_{i=1}^n Nu(\phi_i)={0}##. The attempt at a solution. For ##→##: Let ##\{v_1,...,v_n\}## be...
  15. N

    MHB Finding the Point of Intersection: Is (-2, -2.5) the Solution?

    Hi, I am looking for some help on how you find the point of intersection of the following two lines: 4y = x - 8 2y = 3x + 1 Thanks for any help. /Nichola
  16. S

    Point of intersection between a parabola and a circle

    Homework Statement Sketch the curve C defined parametrically by ##x=t^{2} -2, y=t## Write down the Cartesian equation of the circle with center as the origin and radius ##r##. Show that this circle meets the curve C at points whose parameter ##t## satisfies the equation ##t^{4} -3t^{2}...
  17. MarkFL

    MHB Find Intersections of y=sin(x) and y=1-x^2 | Mangoqueen54

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  18. M

    Point of intersection of tangent line with another line

    Homework Statement Find the pt. at which the tangent line to the curve x=3t^2 - t, y=2t+t^3 at t=1 intersects the line y=2-x. Homework Equations Possibly <6t-1, 2+3t^2> if the tangent is not already present The Attempt at a Solution I am confused about how to go about solving...
  19. F

    MHB Intersection multiplicity

    Define the multiplicity of $f$ at $p$ and the interesction multiplicity of $f,g$ at $p$. Let curves $A$ and $B$ be defined by $x^2-3x+y^2=0$ and $x^2-6x+10y^2=0$. Find the intersection multiplicities of all points of intersection of $A$ and $B$. If we let $f=x^2-3x+y^2$ and $g=x^2-6x+10y^2$...
  20. N

    Help required w/ vectors: general equations, intersection points

    Homework Statement In this question we consider the following six points in R3: A(0,10,3) B(4,18,5) C(1,1,1) D(1,0,1) E(0,1,3) F(2,6,2) a) Find a vector equation for the line through the points A and b b) Find general equations for the line from a c) Find a vector equation for the...
  21. C

    MHB Undirected graph proof, Set intersection

    Hello all, I'm a bit stumped when it comes to formal proofs. I PART A: "Let A,B ⊆{1,2...n} be two sets with A,B > n/2. Prove that the intersection of A ∩ B is nonempty." This part I used contradiction, but didn't get everything. I assumed that if the intersection of A and B was empty, then A∪B...
  22. A

    So, no matter what, if | a-b | is less than every positive ε, then a = b.

    Homework Statement Let \Lambda = N and set A_{j} = [j, \infty) for j\in N Then j=1 to \infty \bigcap A_{j} = empty set Explanation: x\in j=1 to \infty \bigcap provided that x belongs to every A_{j}. This means that x satisfies j <= x <= j+1, \forall j\inN. But clearly this...
  23. S

    Linear Velocity and Angular Velocity Point of Intersection

    So, I have been studying angular velocity and linear velocity--and I want to use this information determine if a ray intersects a plane. linear velocity = dp/dt angular velocity = dΘ/dt thus for linear velocity, you have a point in space: the intersection point could be described as I...
  24. J

    Intersection of a 45 degree angle and an ellipse

    If you are looking at the upper right quadrant of an ellipse centered at (0,0), with a=1 and b=.6, and there is a 45 degree line drawn from (1,.6), how would I find the (x,y) coordinate where the line crosses the ellipse? (I have been out of school for a long time, this is not homework).
  25. maistral

    Intersection of an equation and discrete points.

    How do you calculate the intersection of discrete data points and an equation? Actually I have two ways already, one is to just take the equation of the discrete points then solve it using a root-finding technique. The other would be substituting the x values of the discretized points to the...
  26. B

    MHB Union & Intersection questions

    Sixty six cats signed up for the contest MISS CAT 2013. After the first round 21 cats were eliminated because they failed to catch a mouse. Of the remaining cats, 27 had stripes and 32 had one black ear. All striped cats with one black ear got to the final. What is the minimum number of...
  27. E

    MHB Finding Basis for Intersection of Subspaces of $\Bbb R^n$

    I am looking for a method of finding a basis of the union and intersection of two subspaces of $\Bbb R^n$. My question is primarily about the intersection. Suppose that the basis of $L_1$ is $\mathcal{A}=(a_1,\dots,a_k)$ and the basis of $L_2$ is $\mathcal{B}=(b_1,\dots,b_l)$. Then $v\in L_1\cap...
  28. A

    Intersection coordinates in lattice

    On the drawing below is a hexagonal lattice. For the basis vectors one can choose either the set of arrows in black or the set in yellow. The intersection coordinates of the plane in green seems to be the same regardless of choosing the black or the yellow basis. Why is that? My teacher said it...
  29. C

    The curve formed by the intersection of paraboloid and ellipsoid

    I will state the specifics to this problem if necessary. I need to find the parametric equations for the the tan line at point, P(x1,y1,z1) on the curve formed from paraboloid intersection with ellipsoid. The parametric equations for the level surfaces that make up paraboloid and ellipsoid...
  30. Onionknight

    Intersection of a Quadric Surface and a Plane in 3-D

    1. "Find the equation that describes the intersection of the quadric given by x^2 + y^2 = 4 with the plane x + y + z = 1." 2. Parametric equations for elliptic curve: x = a cos(t) , y = b sin(t) , z = ? 3. Surface is an [EDIT: right circular] cylinder. Plane is not parallel to xy...
  31. Raerin

    MHB How to find the intersection point between two lines

    How to find the intersection point between two lines? line 1: r = (3,1,-1) + s(1,2,3) line 2: r = (2,5,0) + s(1,-1,1)
  32. S

    MHB How Do You Calculate the Volume of a Rotated Solid?

    Question: Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. x = 2\sqrt{y}, x = 0, y = 9 So I know what the graph looks like. But how do i find the points of...
  33. J

    Calculating the Intersection of Subspaces in Vector Spaces

    Given two subspaces of the vector space of all polynomials of at most degree 3 what is the general method to calculate the intersection of the two subspaces?
  34. P

    Finding the Intersection of a Line and Plane

    Say you have a point on one surface. You know the normal vector of the surface at this point. You have a triangle somewhere else in space defined by it's three vertices. How do you find the intersection - if any - between the normal vector at the point on the surface with the triangle? I...
  35. C

    MHB Intersection Points & Finding Unknown Variable

    The line with equation y = x + k, where k is a real number, intersects the parabola with equation y = x^2 + x − 2 in two distinct points if I first made the equations equal each other x + k = x^2 + x − 2 0 = x^2 -2 -k From here i thought you use the discriminate a=1 b=o c=-2-k but this...
  36. bigfooted

    Intersection of straight line with (lagrange) polynomial

    Hi, To calculate the intersection of two straight lines the cross product of the line vectors can be used, i.e. when the lines start in points p and q, and have direction vectors r and s, then if the cross product r x s is nonzero, the intersection point is q+us, and can be found from...
  37. Saitama

    MHB Intersection of line with ellipse - given difference of eccentric angles

    Problem: Find the condition so that the line px+qy=r intersects the ellipse $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ in points whose eccentric angles differ by $\frac{\pi}{4}$. Attempt: Let the points on ellipse be $(a\cos\theta,b\sin\theta)$ and...
  38. G

    Intersection of TWO QUADRICS/Conics

    Homework Statement Find all the plane (x,y) all points of intersection of two quadric: 2x^2-xy+3y^2=36, 3x^2-4xy+5y^2=36 Homework Equations The Attempt at a Solution I want to know the general process to solve something like this. Is the problem solved by using det...
  39. F

    MHB Sets: A,B,C - Intersection & Difference

    Am I right in thinking that if we have 3 sets A,B,C, then with A intersect B represented as AB, we have: A(B\C)=(AB)\C=B(A\C)?
  40. T

    Intersection of two subgroups trivial, union is the whole group

    Homework Statement Let ##G## be a group of order ##n## where ##n## is an odd squarefree prime (that is, ##n=p_1p_2\cdots p_r## where ##p_i## is an odd prime that appears only once, each ##p_i## distinct). Let ##N## be normal in ##G##. If I have that ##|G/N|=p_j## for some prime in the prime...
  41. M

    Set theory: find the intersection

    Homework Statement In a group of 30 people each person twice read a book from books A, B, C. 23 people read book A, 12 read book B and 23 read book C. (a) How many people read books A and B? (b) How many people read books A and C? (c) How many people read books B and C? Homework...
  42. M

    Intersection nested, closed sequence of intervals

    Homework Statement . Let ##\{I_n\}_{n \in \mathbb N}## be a sequence of closed nested intervals and for each ##n \in \mathbb N## let ##\alpha_n## be the length of ##I_n##. Prove that ##lim_{n \to \infty}\alpha_n## exists and prove that if ##L=lim_{n \to \infty}\alpha_n>0##, then ##\bigcap_{n...
  43. P

    Intersection of Two Curves: Do They Meet?

    Homework Statement Show that these curves do not intersect. z=(1/a)(a-y)^2 y^2+z^2=a^2/4 Where a is the radius of the circle and other shape. Homework Equations There aren't any. The Attempt at a Solution I tried setting them equal to each other but got the equation...
  44. D

    Probability of Empty Intersection of Randomly Chosen Planes?

    Let x \in \{-1, 1\}^n and let p(x) = \{w \in \mathbb{R}^n : x \cdot w > 1\}. What is the probability that p(x_1) \cap \ldots \cap p(x_{n+1}) = \emptyset given that x_i are chosen uniformly at random?
  45. R

    Edge dislocation intersection from point defects

    Homework Statement Two edge dislocations having an equal, but opposite in sign, burgers vector are gliding on parallel (111) planes in copper (FCC). Calculate the number of point defects required to bring the two dislocations together. The vertical separation between the dislocations is 1...
  46. M

    Curve Intersection of Surfaces in 3D: Solving for the Parametric Equations

    Homework Statement Show that x=sin(t),y=cos⁡(t),z=sin^2 (t) is the curve of intersection of the surfaces z=x^2 and x^2+y^2=1. Homework Equations I don't think there aren't really any equations relevant for this maybe except the unit circle..? The Attempt at a Solution...
  47. C

    Finding the Vector Equation for the Intersection of Two Planes

    Homework Statement Find the vector equation for the line of intersection of the planes 4x+3y−3z=−5 and 4x+z=5 r = < _, _, 0> + t<3, _, _> Fill in the blanks for the vector equation. Homework Equations The Attempt at a Solution I used the method of elimination of linear...
  48. B

    Proving the Intersection of Subgroups is a Subgroup: A Simple Solution

    Homework Statement Prove that the intersection of any collection of subgroups of a group is again a subgroup Homework Equations The Attempt at a Solution Fixed proof Let H_1 and H_2 be subgroups on G. We first see if H_1 \cap H_2 is again a subgroup. We see if a,b\in H_1 \cap...
  49. C

    Parametric Equation for Intersection of Parabola and Ellipsoid

    I need to find a parametric equation of the vector function which is the intersection of y = x2 and x2 + 4y2 + 4z2=16. I know the graph of the first equation is a parabola which stretches from negative infinity to infinity in the z direction. I also know that the second equation is that of an...
  50. J

    Finding the Intersection of Two Graphs: Exact Solutions and Integrals

    Homework Statement Hi I'm trying to find where these two graphs intersect I would like it to be exact but it isn't quite working. If I have y = tan(x) and y = x^1/3 how can I solve exactly? Homework Equations tan(x) = x^1/3 ? Hmm. I'm not sure. I don't want arctan popping up on the...
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