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. G

    Intersection of ellipses and equivalent problems

    Does anyone know how to determine whether two ellipses intersect? I don't need the precise points but rather only the answer whether there are points. All my attempts led to 4th order polynomials, which are heavy to solve, but considering that I don't need the actual points I assume there must...
  2. Telemachus

    Intersection between the solid and a plane perpendicular

    Homework Statement A solid has a circular base of radius r=4. Find in each case the volume of the solid if all intersection between the solid and a plane perpendicular to a set diameter is: a) A square b) A triangle rectangle isosceles which one of its sides is on the circular base. The...
  3. T

    Finite Intersection Property Question

    I'm having a little trouble distinguishing the line between what the f.i.p implies and what it does not. **EDIT2** Hopefully this will make things more clear What I'm really interested in is a formal definition of the f.i.p regardless of the set in question or the field. Given the sequence...
  4. J

    Finding the Intersection of Two Planes: A Quick and Efficient Method

    Homework Statement P1 = 3x+4y=1 P2 = x +y -z=3 Find the intersection. Homework Equations x = x0 +at y = y0 + bt z = z0 + ct The Attempt at a Solution I used a quicker method. I first did P1 x P2 and got its cross product <-4,3,-1> Then I followed what our teacher had shown us in class -...
  5. C

    Finding Normal Vector to Plane at Intersection Point for HeNe Laser Beam Path

    I'm working on tracing the beam path of a HeNe laser through two prisms, and I'm stuck on trying to find the incident plane in which to use Snell's law. Basically I have the equation of the angled surface of the prism, and I have the point where the beam intersects that plane. Now I need the...
  6. N

    Intersection test for ray tracing

    I know that intersection tests can typically (for hit detection in games) be computed in log n time with reasonable accuracy, but there is a problem that I see with using hierarchical bounding volumes in ray tracing. For example, a ray may intersect a bounding volume however due to the shape...
  7. B

    Intersection of Connected Sets: True or False?

    I have been asked if the following is true or false the intersection of two connected sets is connected ? I would have thought that if their intersection was empty they wouldn't be connected. If they were disjoint or course it would be empty. any ideas?
  8. S

    Area in polar (stuck at the intersection points)

    Homework Statement find the area inside both of the curves r = 4 cos@ r = 2+2cos@ @ = theta Homework Equations ------- The Attempt at a Solution i will say 4cos@ = 2+2cos@ to find the intersection points 4cos@ = 2+2cos@ 2cos@ = 2 cos@ = 1 @ = 0 ! I need the other points!
  9. M

    Intersection of an infinite set

    Let G_{n} = (-1/n,1/n) for all n in N let G= \bigcap^{\inft}_{n=1}
  10. C

    Probability and average area of intersection of rectangles and ellipse

    Hello all, I am neither a physicist nor a mathematician, I am an archaeologist trying to develop a mathematical model for archaeological site detection. The problem is set up like this (and hopefully this will make some sense): The detector moves in parallel transects that can be of any...
  11. K

    Intersection of two vector-valued functions

    Homework Statement Prove that r1 and r2 intersect at (1,1,3). Let r1 and r2 be defined as: r1(t)=t^2i+tj+3t^3k r2(t)=(t−1)i+(1/4)t^2j+(5−t)k Homework Equations Intersection is derived from r1=r2. The Attempt at a Solution I only formally get to this point and then start...
  12. S

    You now have three points that define a plane.

    Homework Statement Find the equation of the plane that passes through the line of intersection of the planes 4x - 3y - z - 1 = 0 and 2x + 4y + z - 5 = 0 and passes through A(1, -3, 2). Homework Equations N/A The Attempt at a Solution I have no clue on how to start this question...
  13. H

    Intersection of two gaussian distribution functions

    Can you please give me formulas which give intersection of two gaussian function f(x, mu, sigma) = 1/sqrt(2*pi*sigma^2) * exp(-(x-mu)^2 / (2*sigma^2)) for the case variances are different. (Note: I think it is time I learn how to use "tex" tags, do you know a good tutorial?)
  14. R

    Slope of the tangent line of an intersection - Directional Derivatives

    Homework Statement Find the slope of the tangent line to the curve of intersection of the vertical plane x - y + 1 =0 and the surface z = x2+y2 at the point (1, 2, 5) Homework Equations Gradients, Cross products The Attempt at a Solution I'm pretty lost here. I think I have to...
  15. M

    Intersecting yz-plane: x^2 + y^2 - 4 Curve

    Homework Statement Does the surface z = x^2 + y^2 - 4 * intersects the yz -plane? If so, find the equation of the curve and write down the points of intersection. The Attempt at a Solution yz-plane, so x=0 1) * becomes z = y^2 - 4 and this is the equation of the curve that intersects...
  16. M

    Equation of curves (intersection)

    Homework Statement Let S be the solid region bounded by the paraboloid z = x^2 + y^2 - 4 for z \le 0 , 0 \le x \le \sqrt{4 - y^2} and 0 \le y \le 2 . Find the Cartesian equations of the curves if the surface S intersects the xy,xz , and yz planes. The Attempt at a Solution...
  17. B

    Generic Intersection of non-planar Surfaces in R^4

    Hi, everyone: How do we show that 2 planar surfaces in R^4 intersect at points (possibly empty sets of points, but not in lines, etc.). I am curious to see how we justify the Poincare dual of the intersection form in cohomology being modular, i.e., integer-valued...
  18. E

    Points of intersection to find area inside a region

    Homework Statement Use a dounble integral to find the area of the region inside r = 1+ cos (theta) and outside r = 2sin (theta). sketch region and indecate the points of intersection. I'm confused how to find the points of intersection of these two equations Homework Equations I've...
  19. T

    Prove Intersection of Subgroups of G is Normal Subgroup

    Homework Statement Suppose that a group G has a subgroup of order n. Prove that the intersection of all subgroups of G of order n is a normal subgroup of G. Homework Equations The Attempt at a Solution I know that I need to do the following: Let S be the set of all subgroups of...
  20. A

    Intersection points of two lines in two-space

    Homework Statement Show that the following pairs of lines intersect. Determine the coordinates of the point of intersection. L1: r= (-3,-1) + t(3,4) L2: r= (6,2) + s(3,-2) Homework Equations ? The Attempt at a Solution I know that eventually the two lines will reach the...
  21. A

    Tangent line of curve of intersection

    Homework Statement Find the parametric equations of the line tangent to the curve of intersection of the paraboloid z = x² + y² and the ellipsoid 4x² + y² + z² = 9 at the point ( -1, 1, 2 ). Homework Equations Probable use of the gradient vector (as this is the chapter we are in)...
  22. N

    Points of intersection with polar equations

    Homework Statement I have to find all of the points of intersection of the curves... r2 = sin(2θ) r2 = cos(2θ) The Attempt at a Solution sin(2θ) = cos(2θ) 2sinθcosθ = cos2θ - sin2θ 2sinθcosθ - cos2θ = -sin2θ cosθ(2sinθ - cosθ) = -sin2θ This is where I'm having a problem, I'm...
  23. S

    HelpGiven a common intersection point, create 3 different planes

    Homework Statement Given a common intersection point (3,4,5), find 3 different planes. Homework Equations None The Attempt at a Solution What I did is let a1x+a2y+a3z=a b1x+b2y+b3z=b c1x+c2y+c3z=c 3=Dx/D 4=Dy/D 5=Dz/D I set D=2, therefore Dx=6 Dy=8 Dz=10...
  24. N

    Intersection pts of polar equations

    Homework Statement I have to find the area of the region that lies inside the curves: r = sin(θ) r = sin(2θ) The Attempt at a Solution I'm assuming the first step would be to find the points of intersection so I know WHERE to integrate from/to, so I set the equations equal to each...
  25. R

    Finding the Centre and Intersection of Circle X2 + y2 + 4x – 16y + 18 = 0

    Homework Statement Find the centre of circle X2 + y2 + 4x – 16y + 18 = 0, show it's radius is 5√2 and find the co-ordinates where it is intersected by y = 3x - 6 Homework Equations The Attempt at a Solution X2 + y2 + 4x – 16y + 18 = 0 by completing the square using X2 + 2px = (x...
  26. D

    Point of intersection of two lines

    Homework Statement Find the point of intersection of two lines x = -9 + 5t y = 1 + t z = 10 - 4t and x = -2 -3t y = 5 + 2t z = 5 + 3t Homework Equations N/A The Attempt at a Solution I have read that you should set two of the equations equal to find the value of t, and...
  27. T

    What are the possible solutions for the inequality |2x-1|+|x+2|\geq 4x?

    Homework Statement |2x-1|+|x+2|\geq 4x Homework Equations The Attempt at a Solution For x<-2 , -(2x-1)-(x+2)\geq 4x x\leq -\frac{1}{7} For x\geq \frac{1}{2} 2x-1+x+2\geq 4x x\leq 1 For -2 \leq x < \frac{1}{2} , -(2x-1)+x+2\geq 4x x\leq \frac{3}{5} after...
  28. S

    Dimension of intersection of U and V

    Homework Statement Prove true: For any subspaces U,V of R^n dim(U intersect V) <= min(dim(U), dim(V)) Homework Equations Min(a,b) = the minimum value of A and B The Attempt at a Solution I know this statement is true however I can't quite figure out where to start on how to...
  29. J

    Proving/Disproving: U and V Intersection in Rn

    So for my homework I have to prove (or disprove) this statement: If U, V are two subspaces of Rn then U \cap V \neq \phi. I just want to make sure; \phi is the null set right? The set with nothing in it?
  30. A

    Parametric Equation of a Line from the intersection of two planes

    Homework Statement Find the parametric equation for a line of intersection of these two planes x+2y+3z=0 4x+5y+6z=5 Homework Equations Normal to plane 1= <1,2,3> Normal to plane 2= <4,5,6> The Attempt at a Solution I know the way to do this problem is to take cross product of...
  31. D

    Proof with intersection of subspaces

    Homework Statement Suppose L, M, and N are subspaces of a vector space. (a) Show that the equation L \cap (M+N) = (L \cap M)+(L \cap N) is not necessarily true. (b) Prove that L \cap (M+(L \cap N))=(L \cap M) + (L \cap N) Homework Equations N/A The Attempt at a Solution...
  32. V

    Maximum Speed - T-shaped intersection

    A couple of months ago I was pulled over by a police officer for running a stop sign. The officer claimed I "plowed right through" the stop sign. I honestly had no idea what he was talking about but wasn't about to argue with a cop. I realized after going back to check the intersection after...
  33. L

    How to find the intersection of two vectors

    L1: r(t) = (-5 + 2t)i + (5 + t)j L2: r(t) = (3 + 4t)i + (4 - 8t)j I know that they are perfendicular but how do I go about finding the point of intersection?
  34. T

    Find the intersection of a particle and a plane in R3

    Homework Statement The starting position of a particle in R3 is (1,1,1) and it's traveling with constant velocity (2,-1,1). Where does it hit the plane {(x,y,z)|x - 2y + z = 4}. And find the angle between the path of the particle and the plane. Homework EquationsThe Attempt at a Solution
  35. L

    Intersection of simply connected

    Hi everybody! I have a question, if I have A and B simply connected subspaces of a geodesic space X, what can be said about their intersection? When is it simply connected? Are there rules for this? I need to prove it in a special case, but I am not able to do it and I was wondering if...
  36. H

    Finding the Intersection of Two Planes in Parametric Form

    Homework Statement Consider the intersection between the following two planes given in parametric form: P1 : x = [2, 4. 3] + s1[1, 2, 1] + s2[2, 5, 4] P2 : x = [1, 0, -5] + t1[3, 8, 7] + t2[2, 1, -5] Find the intersection of the two planes as a line in parametric form. Homework...
  37. D

    How Do You Find the Point of Intersection for These Equations?

    Homework Statement Find the point of intersection Homework Equations Equation 1 y=1/5(2)4x-7 Equation 2 y=1/10(1/2)9-3x The Attempt at a Solution Made them equal each other got rid of the 1/5 one one side and it made the 1/10 turn into 1/2 and now i don't know...
  38. H

    MATLAB MATLAB vector intersection help

    i have the following double arrays: 165 7077 165 7045 165 5175 168 7305 168 3487 169 6196 169 10213 170 7048 165 7077 165 7045 168 5175 169 7305 169 3487 170 6196 170 10213 170 7048 how do i obtain the number and percentage of common values in column 2 of array1 and array2...
  39. T

    Sum and intersection of anihalator spaces

    Homework Statement prove that (U\bigcapW)^{\circ}=W^{\circ}+U^{\circ} First prove That (U\bigcapW)^{\circ}\supseteqW^{\circ}+U^{\circ} Take any f\in (U\bigcap W)^{\circ} Then it is easy to see that for any f\in (U\bigcap W) f(v) =0 but since v\in U and v \in W then f \in...
  40. T

    General questions about intersection of subspaces

    Hey Guys. I have some questions about vector spaces, I would really apreciate if somone could read this and let me know if I understand things or not, and if not let me know where I have it wrong. I am having a lot of trouble UNDERSTANDING how to find the intersection of two vector spaced...
  41. H

    Calc III: sphere & plane intersection.

    Consider the intersection of the sphere (x-3)^2+(y+2)^2+(z-1)^2=13 with the plane x+y=0 a) This intersection should be a familiar curve. Describe the curve. b) Substituing x=-y or y=-x into the equation of the sphere does not give this curve. Explain the difference. I know that the...
  42. J

    Need Help Finding Points of Intersection

    Homework Statement y=x-x^3 y=0 Must find intersection points by way of the x-axis Homework Equations The Attempt at a Solution I know you have to set them equal to each other i just do not know what to do from there.
  43. C

    Intersection point between a line with slope m and f(x)

    Homework Statement f(x) = 2x2-5x-12 is given; part a: find derivative of f(x) using first principles, part b: find the rate of change of f(x) at x=1, part c: the points at which the line through (1, -15) with slope m cuts the graph of f(x), part d: the value of m such that the points of...
  44. J

    Angle of intersection: polar versus cartesian

    Is it correct that the angle of intersection of two curves is the same in x,y coordinates as in r,theta coordinates? If so, why is this?
  45. M

    Equation of plane containing the intersection of two planes

    Homework Statement Find an equation of the plane that passes through the point (-1,2,1) and contains the line of intersection of the planes x + y - z = 2 and 2x - y + 3z = 1 Homework Equations Equation of a plane: a(x-x0) + b(y-y0) + c(z-z0) = 0 The Attempt at a Solution n1 =...
  46. J

    Collision at Intersection: Calculating Direction & Distance

    Homework Statement Note the picture... 11. Two cars approach an intersection as shown. Car 1 weighs 4500 lbs and has a speed of 55.0 mi/hr. Car 2 weighs 3750 pounds with a speed of 60.0 mi/hr. They collide in a completely inelastic collision at the intersection. a) Calculate the...
  47. T

    Solve Math Puzzle: Proving Nonemptiness of Intersection

    Here's a math problem that's giving me a head ache (though to some of you it might seem to be quite trivial) r,s\in N^\ast , r+1\leq s ; |A_i|=r, \forall i\in \{1,2,...,s\} the intersection of any r+1 of sets A_i is nonempty [1] prove that \bigcap_{i=1,s} A_i \neq \emptyset [C]...
  48. T

    What Values of K Make the Intersection of Two Graphs Positive?

    the graph of y=kx+3 intersects the graph of y=x^2 +8x at two distinct points for 'k 'equals what? to be honest I do not know where to start
  49. P

    Tangent vector to curve of intersection of 2 surfaces

    Homework Statement Find the tangent vector at the point (1, 1, 2) to the curve of intersection of the surfaces z = x2 + y2 and z = x + y. Homework Equations The Attempt at a Solution I haven't started the problem, because I'm not sure what the first thing to do is. Do I have to parametrize...
  50. P

    Tangent line of the curve of intersection of two surfaces

    How do I find this?
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