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.
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...
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...
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##.
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!
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...
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...
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?
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...
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...
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)...
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...
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...
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.
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...
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...
Since z=0, the only variable that counts is x.
So the solution would be:
$$\frac {f \left(a + \Delta\ x, b \right) - f(a,b)} {\left( \Delta\ x\right)}$$
$\tiny{\textbf{2.4.10}}$
$\begin{array}{rl}
(x+4)^2+(y+11)^2&=169 \\
(x-9)^2+(y+5)^2&=100 \\
(x-4)^2+(y-5)^2&=25
\end{array}$
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...
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: #...
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 $(x-\epsilon, x+\epsilon)\subset U$...
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...
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...
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...
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...
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?)...
$\textbf{xy-plane}$ 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...
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...
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.
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.
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...
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...
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...
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 +...
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...
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...
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...
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...
## 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...
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...
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...
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.
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...
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 )...
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.
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...