# 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.

View More On Wikipedia.org
1. ### Calculate the area intersected by a sphere and a rectangular prism

Think of a 3D rectilinear grid made of these rectangular cells, some of the cells will intersect with the sphere. I am trying to compute each intersecting area and the total sum. Ideally the total sum of the intersecting area should be close to ##4 \pi r^2##. I have not found any literature...
2. ### I How to establish which side of a square a ray will intersect?

Consider some ray ## \bar{r} ## that starts at point ## A=(a_x,a_y) ## and faces some direction and consider an upright square ( i.e. it's not rotated ) at some location: Question: if we let the ray continue until hitting the square, how can we detect which face of the square was hit? is there...
3. ### POTW Flat Modules and Intersection

Let ##M## be a flat module over a commutative ring ##A##. Suppose ##X_1## and ##X_2## are submodules of an ##A##-module ##X##. Prove that ##(X_1 \cap X_2) \otimes_A M = (X_1 \otimes_A M) \cap (X_2 \otimes_A M)## as submodules of ##X\otimes_A M##.
4. ### B Exploring the Intersection of Oscillations/Waves & Particles in EM Fields

For an upcoming presentation I am looking for a topic which covers both the field of oscillations/waves and particles in electromagnetic fields. Do you have any interesting ideas for a possible topic? Many thanks for your help in advance!

6. ### Find the coordinates of intersection between tangents and given curve

ooops...this was a bit tricky but anyway my approach; ... ##\dfrac{dy}{dx}=-2x## therefore; ##\dfrac{y-7}{x+1}=-2x## and given that, ##y=4-x^2## then; ##4-x^2-7=-2x^2-2x## ##x^2+2x-3=0## it follows that, ##(x_1,y_1)=(-3,-5)## and ##(x_2,y_2)=(1,3)##. There may be another approach...
7. ### Use Stokes' theorem on intersection of two surfaces

I parameterize surface A as: $$A = (2cos t, 0, 2sin t), t: 0 \rightarrow 2pi$$ Then I get y from surface B: $$y = 2 - x = 2 - 2cos t$$ $$r(t) = (2cost t, 2 - 2cos t, 2sin t)$$ Now I'm asked to integral over the surface, not solve the line integral. So I create a new function to cover the...
8. ### Create curve function from intersection of two surfaces

What I do is set the two equations equal to one another and solve for z. This gives: $$z = \sqrt{x^2+2y^2-4x}$$ which is a surface and not a curve. What am I doing wrong?
9. ### B Decision for conditional probability instead of intersection of events

Hello, I have a question about the following sentence and would appreciate if someone could explain how to read out the conditional probability here. "Each microwave produced at factory A is defective with probability 0.05". I understand the sentence as the intersection ##P(Defect \cap...
10. ### I The number of intersection graphs of ##n## convex sets in the plane

Let ##S## be a set of n geometric objects in the plane. The intersection graph of ##S## is a graph on ##n## vertices that correspond to the objects in ##S##. Two vertices are connected by an edge if and only if the corresponding objects intersect. Show that the number of intersection graphs of...
11. ### B Intersection of line and curve

I have a formula y=log(x)/log(0.9) which has this graph: I want to find the intersection of this curve and a tangent line illustrated in this rough approximation: The axes have very different scales, so the line isn't actually a slope of -1, it's just looks that way. How can I figure out: 1)...
12. ### MHB Proof of a set union and intersection

Hello! Lately, I've been struggling with this assignment. (angle brackets represent closed interval) I figured out that: a) union = R intersection = {0} b) union = (0, 2) intersection = {1} I asked my prof about this and she explained to me that it should be shown that if a set is an...
13. ### Who Has the Right-of-Way in an Intersection Collision?

Watching a court case on TV. This is the set up: Blue car at STOP sign, turning right onto 4-lane road. Red car on main road, but changes lanes in intersection. There is a collision. Who is at-fault? (Or who is more at fault?) I have always understood that it is illegal to change lanes in an...
14. ### Partitioning 5 Rays: Nonempty Intersection

I need to show the following thing: Given a collection of 5 rays (half-lines) in the plane, show that it can be partitioned into two disjoint sets such that the intersection of the convex hulls of these two sets is nonempty.
15. ### Prove that the intersection of subspaces is compact and closed

Given that one of the ##S_i## (let's name it ##S_{compact}##), is compact. Assume there is an open cover ##\mathcal V## of ##S_{compact}##. By definition of a compact subspace, there is a subcover ##\mathcal U## with ##n<\infty## open sets. Notice that ##\forall x\in (\bigcap_i S_i)##, ##x\in...
16. ### Intersection of two line segments from uniform distribution

Hi, I found this question online and made an attempt and would be keen to see whether I am thinking about it in the right manner? Question: Find the probability of two line segment intersecting with each other. The end points of lines are informally sampled from an uniform distribution...
17. ### Intersection of a function f(x,y) with a plane

Since z=0, the only variable that counts is x. So the solution would be: $$\frac {f \left(a + \Delta\ x, b \right) - f(a,b)} {\left( \Delta\ x\right)}$$
18. ### MHB 2.4.10 3 circles one intersection

$\tiny{\textbf{2.4.10}}$ $\begin{array}{rl} (x+4)^2+(y+11)^2&=169 \\ (x-9)^2+(y+5)^2&=100 \\ (x-4)^2+(y-5)^2&=25 \end{array}$ ok i solved this by a lot of steps and got (1,1) as the intersection of all 3 circles these has got to be other options to this. basically I expanded the...
19. ### Span(S) is the intersection of all subspaces of V containing S

Homework Statement:: I want to understand the proof for the following theorem: span(S) is the intersection of all subspaces of V containing S. Relevant Equations:: N/A I know that if ##W## is any subspace of ##V## containing ##S## then ##\text{span}(S) \subseteq W##. I have read (Page 157: #...
20. ### MHB Proving Topology in X: A Look at Union & Intersection

Hey! :giggle: We consider the set $X=\mathbb{R}\cup \{\star\}$, i.e. $X$ consists of $\mathbb{R}$ and an additional point $\star$. We say that $U\subset X$ is open if: (a) For each point $x\in U\cap \mathbb{R}$ there exists an $\epsilon>0$ such that $(x-\epsilon, x+\epsilon)\subset U$...
21. ### Engineering BMD of 4 beams joined at an intersection

4 beams are supported by the column at one side, another side is jointed together at the intersection. I have assigned UDL of 20 kN/m on all the beams. 2 beams are longer (3m), 2 beams are shorter (2m) . Surprisingly, the BMD of the longer beams is hogging at the middle part , while the BMD of...
22. ### Proving intersection of finitely many open sets is open

Define a collection of open sets to be denoted as ##P_i##, ##1\leq i\leq N## where ##N\in \mathbb{Z}^+##. Let ##x\in\cap_{i=1}^N P_i##. By definition, ##x## must belong to every single ##P_i##. In particular, ##x\in P_1## and ##x\in P_2##. Since ##P_1## and ##P_2## are open, there exist...
23. ### I Finding intersection of two algebraic curves

Given two algebraic curves: ##f_1(z,w)=a_0(z)+a_1(z)w+\cdots+a_n(z)w^n=0## ##f_2(z,w)=b_0(z)+b_1(z)w+\cdots+b_k(z)w^k=0## Is there a general, numeric approach to finding where the first curve ##w_1(z)## intersects the second curve ##w_2(z)##? I know for low degree like quadratic or cubics...
24. ### Unable to find the intersection between a circle and ellipse

Given: x^2+xy+y^2=18 x^2+y^2=12 Attempt: (x^2+y^2)+xy=18 12+xy=18 xy=6 y^2=12-x^2 (12)+xy=18 xy=6 Attempt 2: xy=6 x=y/6 y^2/36+(y/6)y+y^2=18 43/36y^2=18 y ≠ root(6) <- should be the answer Edit: Just realized you can't plug the modified equation back into its original self I plugged y=6/x...
25. ### Intersection of a few surfaces

Summary:: Describe what the intersection of the following surfaces - one on one - would look like? Cone, sphere and plane. My answers : (1) A cone intersects a sphere forming a circle. (2) A sphere intersects a plane forming a circle. (3) A plane intersects a cone forming (a pair of?)...
26. ### MHB -gre.ge.04 intersection of parabola and line

$\textbf{xy-plane}$ above shows one of the two points of intersection of the graphs of a linear function and and quadratic function. The shown point of intersection has coordinates $\textbf{(v,w)}$ If the vertex of the graph of the quadratic function is at $\textbf{(4,19)}$, what is the value of...
27. ### MHB Understanding the Intersection of Inductive Sets & the Limits of λ Cardinality

By ZFC, the minimal set satisfying the requirements of the axiom of infinity, is the intersection of all inductive sets. In case that the axiom of infinity is expressed as ∃I (Ø ∈ I ∧ ∀x (x ∈ I ⇒ x ⋃ {x} ∈ I)) the intersection of all inductive sets (let's call it K) is defined as set K = {x...
28. ### Volume Of Intersection Between Square Pyramid And Sphere

I'm assuming the way to go about it is to integrate in spherical coordinates, but I have no idea what the bounds would be since the bottom edges of the square pyramid are some function of r, theta, and phi.
29. ### I Intersection of a 4D line and a 3D polyhedron in 4D

Is the intersection of a 4D line segment and a 3D polyhedron in 4D a point in 4D, if they at all intersect? Intuitively, it looks like so. But I am not sure about it and how to prove it.
30. ### Probability Questions: Union, Intersection and Combinations

Question 1: a) T' is the complementary event of T Therefore, T'=1-T In set T = {3,6,9,12} P(T)=4/12 =1/3 P(T')=1-1/3=2/3 b) The addition rule states; P(A ∪ B)=P(A)+P(B)-P(A⋂B) Therefore, P(S ∪ E) = P(S)+P(E)-P(S⋂E) S={1,4,9} P(S)=3/12=1/4 E={2,4,6,8,10,12} P(E)=6/12=1/2 (S⋂E)={4} P(S⋂E)=1/12...
31. ### Intersection of a tangent of a hyperbola with asymptotes

Summary:: Question: Show that the segment of a tangent to a hyperbola which lies between the asymptotes is bisected at the point of tangency. From what I understand of the solution, I should be getting two values of x for the intersection that should be equivalent but with different signs...
32. ### Intersection of a circle and a parabola

We have a circle (x^2 + y^2=2) and a parabola (x^2=y). We put x^2 = y in the circle equation and we get y^+y-2=0. We get two values of y as y=1 and y=-2. Y=1 gives us two intersection point i.e (1,1) and (-1,1). But y=-2 neither it lie on the circle nor on the parabola. The discriminant of the...
33. ### MHB Find the points of intersection of a line and a circle

How do I algebraically prove how many times the line y=-5 intersects the circle (x-3)^2 + (y+2)^2 =25?

35. ### MHB Find the intersection point for x = -5 + 8t, y = 1 + 10t, z = 9 + 8t ; -2x + 8y + 8z = 10

Find the intersection. x = -5 + 8t, y = 1 + 10t, z = 9 + 8t ; -2x + 8y + 8z = 10
36. ### Find all points of intersection

First I try to visualize it: w = Surface 1, is a spheroid w_2 = Surface 2 is a cone stretching up the z axisThen I calculate their gradients: $$∇w = (8x, 2y, 2z)$$ $$∇w_2 = (2x, 18y, 2z)$$ The points where they intersect at 90 degrees is when dot product is zero. $$∇w \cdot ∇w_2 = 0$$ 16x^2 +...
37. ### I Find the intersection point of an infinite power tower and a primorial

Consider ##f(x) = {^{\infty}x} = x \uparrow \uparrow \infty## and ##g(x)=p_{x}###, where ##p_x### is the primorial function and is defined such that ##p_n### is the product of the first ##n## prime numbers. For example, ##p_{4}### ##= 2×3×5×7=210## Let the point of intersection be defined as...
38. ### Pushing a stalled car out of an intersection

Hints given: -Start with free body diagram. Use the relationship between impulse and momentum to find the final velocity of the car after he has pushed for time t. -Use a kinematic equation to relate the final velocity and time to the distance traveled. -What is his initial velocity? My...
39. ### MHB What is the intersection of two spans?

Hey! :o Let \begin{equation*}v_1:=\begin{pmatrix}1 \\ 2\\ -1 \\ 3\end{pmatrix}, v_2:=\begin{pmatrix}1 \\ 1\\ 1 \\ 1\end{pmatrix}, v_3:=\begin{pmatrix}-1 \\ 1\\ -5 \\ 3\end{pmatrix} , w_1:=\begin{pmatrix}1 \\ 2\\ -3 \\ 3\end{pmatrix}, w_2:=\begin{pmatrix}1 \\ 0\\ 0 \\ 1\end{pmatrix}\in...
40. ### Geometry error: no intersection found in mcnp

Hello All, I have yet another MCNP question. I received the following error "geometry error: no intersection found mcnp" when trying to run a a simulation. I looked at the output and according to it I have an infinite volume in cells 14 and 500. I plotted the geometry and don't see how its...
41. ### I Intersection of a plane with a segment in n dimensions

I take 2 points given by the vectors of coordinates ##\vec{p}_i,\vec{p}_j## and a plane spanned by ##\vec{e}_k,k=1,2##. All the vectors are in dimension n. I want to find the intersection of the segment described by the extremities given by the ##\vec{p}_k## with the plane, if any. Is it...
42. ### Range of f(x): Intersection of h(x) and g(x) Ranges

## Let~~f(x)=h(x)+g(x) , where~~h(x)=10^{\sin x}~~and~~g(x)=10^{\csc x}## ##Then,~~D_f = {D_h}\cap {D_g}## ##Clearly,~~D_h=ℝ~~and~~D_g=ℝ-\{nπ|n∈ℤ\}## ##∴~~D_f =ℝ-\{nπ|n∈ℤ\}## After considering the new domain, the range of ##\sin x## in ##10^{\sin x}## is ##[-1,1]-\{0\}## Therefore, the range of...
43. ### How to find the volume when solids intersect?

I know that to find the volume under a surface and above a boundary we have to integrate twice. I can explain myself with an example :- Lets' consider that we need to find the volume under the surface z = \sqrt{1-x^2} and above the region bounded by y^2 = x and positive x-axis and x=5 ...

Sorry for the really messy work I know I have a problem. The other questions that the problem asked before the one I need help with are as follows: Find the intercepts and sketch the plane. Find the distance between the plane and the point (1,2,3) Find the angle between the plane and the xz...
45. ### MHB Can Three Circles Intersect at a Common Point?

studying with a friend there was the intersection of 3 circles problem which is in common usage here is my overleaf output I was wondering if this could be solved with a matrix in that it has squares in it or is there a standard equation for finding the intersection of 3 circles given the...
46. ### MHB Polar Coordinates Intersection

Determine the polar coordinates of the two points at which the polar curves r=5sin(theta) and r=5cos(theta) intersect. Restrict your answers to r >= 0 and 0 <= theta < 2pi.
47. ### Finding the cardinal number for the intersection of two sets

My Question : 1.Why are the inequalities considered? Why not simply use ##n(A\cap B) = n(A)+ n(B)-n(A\cup B)## to get ## n(A\cap B) = 39## ? 2. The way I interpret this is : If the set for people liking cheese was to be a subset of the set for people who like apples then the most number of...
48. ### I The dimensions of locus that is intersection of loci

It seems to me that for a set of loci of cardinality M having dimensions Di in a space of dimension N, aside from degenerate intersections (e.g., a pair of spheres that touch at a single point), the dimension of the net intersection locus L is: L = N - ∑ ( N - Di ) = ( ∑ Di ) - N ( M - 1 )...
49. ### I Area of the intersection of two regions in the plane

I have two regions, given by ##y>\sqrt{2}x - \frac{1}{4x}## and ##y< \sqrt{2}x + \frac{1}{4x}##. How can I find the area of their intersection? If their is no easy analytical way, could someone perhaps use a computer? I am not sure how.
50. ### Describing an object made by the intersection of 2 surfaces

Homework Statement Describe and sketch the geometric objects represented by the systems of equations Homework Equations x2 + y2 + z2 = 4 x + y + z = 1 The Attempt at a Solution I can sketch both objects: 1) sphere with center (0,0,0) and radius 2 2) "simple" plane with intersection...