What is straight lines: Definition and 44 Discussions
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist embedded in two, three, or higher dimension spaces. The word line may also refer to a line segment in everyday life, which has two points to denote its ends (endpoints). A line can be referred to by two points that lie on it (e.g.
A
B
↔
{\textstyle {\overleftrightarrow {AB}}}
) or by a single letter (e.g.
ℓ
{\displaystyle \ell }
).
Euclid described a line as "breadthless length" which "lies evenly with respect to the points on itself"; he introduced several postulates as basic unprovable properties from which he constructed all of geometry. Euclidean line and Euclidean geometry are terms introduced to avoid confusion with generalizations introduced since the end of the 19th century, such as non-Euclidean, projective and affine geometry.
Hi, i'm trying to solve this problem.
It's exercise 3 on page 5 from this book:
Challenging mathematical problem with elementary solutions
The solution is on page 41:
I'm OK with the 4 circles in case 1: i can pick (inside/outside):
ABC + D,
ABD + C,
ADC + B,
BCD + A.
What i cannot...
In the universe do straight lines exist? I know over long distances like interstellar and even shorter distances like between the Earth and around the gravity of the moon lines tend to curve, but do straight lines exist anywhere? Or just a desire for them to exist in nature if not for gravity...
I think that we can say that PPR = α*PRPS
where PR and PS are the points where occurs the intersection on the line R and S.
Obs: line r and s are found by knowing that the straight line intersection of two planes are
n1 X n2 [cross product]
Lr = (0,1,-2) + y(-1,1,1)
Ls = (0,1,-1) + u(1,2,1)...
While determining the condition for the pair of straight line equation
##ax^2+2hxy+by^2+2gx+2fy+c=0##
or ##ax2+2(hy+g)x+(by^2+2fy+c)=0 ## (quadratic in x)
##x = \frac{-2(hy+g)}{2a} ± \frac{√((hy+g)^2-a(by^2+2fy+c))}{2a}##
The terms inside square root need to be a perfect square and it is...
Hello,
I want to see word problems on pair of straight lines to know the real-time applications.
I want to find out what it means if a straight line intersects/is parallel to other straight line.
Thanks!
Hello! Please help me with this question.
I'm currently studying Light and Geometric Optics and have become quite confused.
When looking at diagrams of mirrors, and light relfection into eyes, why is it that eyes only see things when tracing the light in straight lines? For example, if an...
Homework Statement
Find the angle between the straight lines:
##(x^2+y^2)(\cos^2{\theta} \sin^2{\alpha} + \sin^2{\theta})=(x \tan{\theta} - y \sin{\alpha})^2##
Homework Equations
[Not applicable]
The Attempt at a Solution
Dividing by ##x^2##,
## (1+(\frac{y}{x})^2)(\cos^2{\theta}...
Hello,
I VAGUELY recall reading, some many years ago, a statement to the following...
"The Greeks were obsessed with circles. Had they relaxed this obsession, they may have seen the significance of modeling curves with small straight lines, and thereby anticipated the Calculus."
Is there any...
Here's something I've been wondering about: does non-Euclidean modern physics imply that there are no straight lines in our universe? If so, how is this possible? With any circular object or space, one can always draw a straight line through it, right? Thanks.
Homework Statement
Air is flowing with a speed v in the direction (-1, -1, 1,) calculate the volume of air flowing through the loop consisting of straight lines joining (in order i presume) (1,1,0) (1,0,0) (0,0,0) (0,1,1) (1,1,0)
Homework EquationsThe Attempt at a Solution
I assume you have to...
Homework Statement
Hello,
I have a quick question about the following problem
F = (2y+3)i+xzj+(yz-x)k
and straight lines from (0,0,0) to (0,0,1) to (0,1,1) to (2,1,1)
Considering C1 is the line from (0,0,0) to (0,0,1)
C2 is the line from (0,0,1) to (0,1,1)
and C3 is the line from (0,1,1) to...
Homework Statement
Find the angle between two straight lines,##y=x## and ##y=2x##.
Find the general formula for finding the angles.
Homework Equations
##f(x)=mx+c##
##\tan(\theta)=\frac{opposite}{adjacent}##
The Attempt at a Solution
First I assign a value for x and calculate the...
Homework Statement
If all the lines given by the equation (3\sin \theta + 5\cos \theta )x+(7\sin \theta - 3\cos \theta )y+11(\sin \theta - \cos \theta)=0 pass through a fixed point (a,b) forall theta in R then |a-b|=
Homework Equations
The Attempt at a Solution
Dividing both sides by...
Suppose I have a parameterized line ##\phi:\mathbb{R}\to\mathbb{R}^n## given by ##\phi(t) = (x^\mu(t))|_{\mu=1}^n##. How can I tell that the line is straight.
My best answer so far is that at every time ##t## the acceleration (2nd derivative) is parallel to the velocity (1st derivative), i.e...
Show that one of the bisectors of the angles between the pair of straight lines ax2+2hxy+by2=0 will pass through the point of intersection of the straight lines ax2+2hxy+by2+2gx+2fy+c=0 if
h(g2-f2)=fg(a-b)
Please help
If each of the equations ax^2+2hxy+by^2+2gx+2fy+c=0 and ax^2+2hxy+by^2-2gx-2fy+c=0 represents a pair of straight lines , find the area of the parallelogram enclosed by them .
Please help
Linearity vs. "takes straight lines to straight lines"
Homework Statement
Prove that if ##\Lambda:\mathbb R^n\to\mathbb R^n## is a bijection that takes straight lines to straight lines, and is such that ##\Lambda(0)=0##, then ##\Lambda## is linear.
Homework Equations
Fock's theorem implies...
Homework Statement
Lines li: ix+(i+1)y+(i+2)=0; i=1,2 intersect at___ point?
Homework Equations
The equation is required at first!
The Attempt at a Solution
I am confused!
How and where should I replace the values of i with 1 or 2?
This is the first step but i am unable to do...
1.There is a circle with the equation x^2 + y^2 - 2ax = 0. A line is drawn through the centre of the circle which is parallel to the line x+2y=0. and also intersects the circle at A and B. Find the area of the triangle AOB.
My attempt-
I calculated the slope of the given line(-1/2).So the...
1. A point P(x,y) is given equidistant from the points A(a+b,b-a) and B(a-b,a+b), then prove that bx = ay
also find the locus of the variable point Z(a cos (theta), b sin (theta)), where (theta) is a variable quantity.
2. T0 prove that ax = by
3. In an attempt towards...
Homework Statement
Determine whether the following two lines intersect:
(x-2)/2 = (y+3)/1 = (z-4)/-3 ,and (x+3)/4 = (y+4)/1 = (-z+8)/4
Find an intersection point, then find the distance between the lines.
Homework Equations
Symmetric equations of a...
Homework Statement
If the distance of any point (x,y) from the origin is defined as
d (x, y) = max {|x|,|y|},
d (x, y) = a non zero constant, then the locus is
(a) a circle
(b) a straight line
(c) a square
(d) a triangle
Homework Equations
The Attempt at a Solution
I don't...
I tried asking a similar question earlier, but based on the answers I got I wasn't able to convey myself well.
Here's Feynman's argument (at least as I understand it) for why, from a Quantum perspective, light (roughly) travels in straight lines from points A to B:
1) the photon will travel...
Hi to everyone.
I'm developing a puzzle game which involves some concepts of geometry.
Suppose you have a rectangle formed by points A, B, C and D. Now, I add points X1 (lies on line AB), X2 (inside the rectangle) and X3 (lies on line AC).
I'd like the output to be the two rectangles...
Homework Statement
Let \gamma be a stright line in a surface M. Prove \gamma is a geodeisc
The Attempt at a Solution
In a plane we know a straight line is the shortest distance between two point. I am not sure if this applies to straight lines on a surface.
Further more, there...
If a photon is detected at point B having been emitted at point A, can one deduce with near certainty that it must have traveled continuously in a straight line from A to B at velocity c?
Homework Statement
http://img193.imageshack.us/img193/1066/65157866.png
The Attempt at a Solution
Since v is a constant vector, I took it out the front of the integral and used the ftc so the integral of the derivative collapses to γ(t) with terminals a to b. Evaluating leads to γ(b) -...
I've been thinking about this for some time now and I wanted to get some feedback on it. If photons are supposed to travel along straight lines (for the sake of simplicity let's neglect the curvature of space-time), then how could one photon possibly be used in a double slit experiment? More...
OK so I'm familiar with the idea that EM radiation propagates through space in straight lines at a uniform speed, that of light.
The idea that light moves in waves, that each wavelength carries a uniform energy and that the wavelength differs yet because the same speed is maintained each...
This is my second post on PF =)
I want to ask what is the use of Families of straight lines?
I am thinking of A Family of Straight Lines Passing Through the Intersection of Two Lines.
We have the equation: L1+kL2=0 where L1=L2=0 and k is a variable, right?
But is it said that L2 is not...
Homework Statement
Problem from Arnold's "Mathematical Methods of Classical Mechanics" on page 59.
Find the differential equation for the family of all straight lines in the plane in polar coordinates.
Homework Equations
\Phi=\displaystyle\int^{t_2}_{t_1}...
can anyone give me a detailed explanation on how to derive equation for a straight line, which is made up of points, each point representing a complex number..//
pls help
Given the real function f(x,y) = x^2 - 4xy + 3y^2, the equation f(x,y) = 0 shows in a graph as 2 straight lines, y=x and y=x/3. For pairs (x,y) between the lines, f(x,y) < 0; for (x,y) outside the lines, f(x,y) > 0.
It is easy to prove the above, by substituting y=mx in the equation f(x,y)=0...
Homework Statement
How do you show that in spectra taken along the equator of a rotating rigid sphere the spectral lines are tilted but straight?
Homework Equations
There were no equations given in the problem, so I guess you can use whatever equations you know.
The Attempt at a...
How many times per day (00:00-24:00) do the hour and minute hand form a straight line together? Would this be the same for when they overlap? Say, the minute hand goes pi/1 hr and the hour hand goes (pi/12)/1 hr. Then what?
On R^n, I'd say the only smooth transformations taking straight lines to straight lines are the affine transformations.
Would I be right saying that?:smile:
How would one go about proving that?
The explanation I've heard for gravity as depicted by relativity, is that it's the consequence of mass curving spacetime. This means that the reason why the light from a star behind Sun is bent is because evening though the light is traveling a straight line, since space itself is bent...
I'm trying to solve this contour integral shown on the attached file, I know usually that they involve curved lines. I know that this is trivial but I need some help with the problem. Please take a look.
Does straight lines have slopes of tangent?
Does straight lines have slopes of tangent?.. :confused: i had no clue when this question came on a test..so i just left it...do they?..i just thought maybe they didn't since they have no limit. Hope i get an answer for this question.
Tanya
Why thunderbolts don't travel in straight lines ?
I think the electrical charge likes to move in the shortest path to the Earth and the shortest path must be a straight line !