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. W

    When Do Lines in Space Not Intersect?

    Hi, I need some help with this question: Find all values of \ k for which the lines do not intersect. \ (x-2,y+1,z-3) = (r,0,3r)\ and\ (x,y,z) = (2,1,4)\ +\ s(2,k,6) I put the first equation in vector form: \ (x,y,z) = (2,-1,3)\ +\ r(1,0,3) Now I know that if the direction...
  2. P

    Find Line Intersection: Symmetric Equations

    Find symmetric equations of the line that passes through the point (0,1,2) and meets each of the lines x = y = z + 2 and x/-2 = (y+3)/1 = z/3. Those equations can be written as: r = (0, 0, 2) + t(1, 1, 1) r = (0, -3, 0) + s(-2, 1, 3) Now, I can't seem to find any direction to go with...
  3. B

    How Can Two Equations Have Multiple Points of Intersection?

    i have y=x^2-x and y=x from this x^2-x=x therefore: x^2=0, and x then equals zero. putting this info into y=x, y=0 this gives the points (0,0). however, in the answer book, it shows that the points of intersection are (0,0) and (2,2). how is it possible to do this?! appreciate any...
  4. D

    Vector function for the curve of intersection of the paraboloid

    Original question: a) Find a vector function for the curve of intersection of the paraboloid z = 3x^2 + 2y^2 and the cylinder y = x^2. b) Show that this curve passes through (1,1,5) but not (3,3,9). I really have no idea how to do either parts of this question. Any help would be greatly...
  5. R

    Finding the Intersection Point of Two Lines in Vector Form

    I'm using the following equation to represent lines (x, y) = (start_x, start_y) + t((end_x, end_y) - (start_x, start_y)) I'm trying to find the interesection point of two lines written in this form. I have been able to solve for t and plug it back into the equation, but i get two values...
  6. O

    Intersection of surfaces/tangent vector

    Hi, how do I find a tangent vector to the curve intersection of the surfaces 2x^2+2y^2-z^2=25 and x^2+y^2=z^2 which has positive x-direction? Thanks in advance.
  7. Oxymoron

    Intersection of disjoint SETS is empty

    The question is: Suppose W and X are subspaces of R^8. Show that if dimW=3, dimX=5, and W+X = R^8, then W \cap X = \{0\}. I can see this is obvious iff W and X are disjoint sets. If we add members of W to X in the usual way, and we get the new set W+X whose dimension is now 8 (given), then...
  8. T

    Basis for the intersection of two spans

    Let S and T be two spans of vectors, what's the general method to find a basis for the intersection of S and T (SnT)? Thanks
  9. L

    2 N-Dimensional Space Intersection

    ok... first of all, I was discussing with my friend, he propose an argument of, if you have two N-Dimensional Spaces, (let's called it Sn1 and Sn2), they will form another M-Dimensional Spaces, which M is either 0 or M bigger or equal one smaller or equal N-1... what he said was that if Sn1...
  10. T

    Finding the Line of Intersection for Two Planes

    Hello, How do I find the line of intersections of the two planes 7x - 2y + 3z = -2 and -3x + y + 2z + 5 =0, without having to resort to solving it by row reduction?
  11. E

    How Do I Find the Intersection of \sin(x) and \cos(x)?

    how do i find intersection of sin(x) and cos(x)? wat method do i use?
  12. M

    Intersection of planes and lines in space

    Could anyone help me summarize or if anyone knows good tricks in solving problems of lines intersecting with planes, etc, in 2d or 3d, the concept is same, but just want others opinion on what its basic idea is. i am able to do problems but i don't really understand them, i have to go back in my...
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