What is Straight line: Definition and 241 Discussions
In geometry, the notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth. Lines are an idealization of such objects, which are often described in terms of two points (e.g.,
A
B
↔
{\displaystyle {\overleftrightarrow {AB}}}
) or referred to using a single letter (e.g.,
ℓ
{\displaystyle \ell }
).Until the 17th century, lines were defined as the "[...] first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which [...] will leave from its imaginary moving some vestige in length, exempt of any width. [...] The straight line is that which is equally extended between its points."Euclid described a line as "breadthless length" which "lies equally with respect to the points on itself"; he introduced several postulates as basic unprovable properties from which he constructed all of geometry, which is now called Euclidean geometry to avoid confusion with other geometries which have been introduced since the end of the 19th century (such as non-Euclidean, projective and affine geometry).
In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry, a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation, but in a more abstract setting, such as incidence geometry, a line may be an independent object, distinct from the set of points which lie on it.
When a geometry is described by a set of axioms, the notion of a line is usually left undefined (a so-called primitive object). The properties of lines are then determined by the axioms which refer to them. One advantage to this approach is the flexibility it gives to users of the geometry. Thus in differential geometry, a line may be interpreted as a geodesic (shortest path between points), while in some projective geometries, a line is a 2-dimensional vector space (all linear combinations of two independent vectors). This flexibility also extends beyond mathematics and, for example, permits physicists to think of the path of a light ray as being a line.
I would like to use the Calculus of Variations to show the minimum path connecting two points is a straight line, but I wish to do it from scratch without using the pre-packaged general result, because I'm having some trouble following it.
Points are ##(x_1,y_1),(x_2,y_2)##.
And we are to...
I stumbled upon this video on YouTube:
Here is a screenshot with some colored lines added of the part that generated a few questions in my head that I hope some of you smart folk can answer for me.
In the video, the spherical magnet (at the end near the two big magnets) appears to increase...
I'm assuming the boat is traveling north at an 8 m/s and the river is flowing east at a 2 m/s. For the boat to move in a straight line, it has to aim at an upstream angle given by #theta#.
Using SOH CAH TOA, ##v_r = 2m/s##, ##v_b = 8m/s##
$$\theta =\sin^{-1} \frac{v_r}{v_b}$$
$$\theta = 14...
Can we say,
(y + z ) x1 = (y1 + z1) x is also an equation of a straight line in 3 dimensional space,
where (x1,y1,z1) and (x,y,z) are the coordinates of a given point and a variable point respectively on a 3D line that passes through the origin,
have seen equation of a straight line in 3...
t=0 => v(0) = 4(0) - 3(0)^2 = 0m/s
t=2 => v(2) = 4(2) - 3(2)^2 = -4m/s
Vavg => (v(0) + v(2))/2 = -2m/s
When researching the answer, I noticed that they used integrals to solve this question. The only problem is that we never learned about integrals/ derivatives or anti derivative. Is there any...
Hi All,
I am trying to solve what I think should be a simple problem, but I must be missing something because I am struggle to solve it. The situation is shown below:
So to summarise:
A body starts at Point A
It moves in a straight line to Point B, covering a distance of 10m
The time taken...
Visualize a lid to a toddler snack cup. Making an entry into an object which is to be placed in nature which can be used by humans but not easily by animals, would the curves used in some toddler snack cups be more durable and flexible without being floppy, or would straight line radii from the...
Find the question here; My interest is on question ##3(c)## only.
My approach, Let the co ordinates of ##C##= ##(x,y)## then considering points ##B## and ##C##. We shall have the gradient given by;
##\dfrac {y-4}{x-1}##=##-2##
also from straight line equation, considering points ##A## and...
This is the question ... I have it's solution ...
My problem : I can't understand why dx=R/cos^2(teta) dteta
I have thought many hours but I couldn't find it's reason ... Can anyone please help with this ?!
Find the problem below;
My approach,
if##x=0##, then ##y=b## and if ##y=0##, then ##x=a##, therefore our co-ordinates are ##(a,0)## and ##(0,b)##. The gradient will give us,
$$-3=\frac {b-0}{0-a}$$
It follows that, ##b=3a##, therefore
$$20=\sqrt {(3a-0)^2+(0-a)^2}$$
$$400=9a^2+a^2$$
##a=2\sqrt...
Find the question here ( this one is a pretty easy question).
I have attempted this in the past using varied approach, this is in reference to part(b) of the question... i have previously used pythagoras theorem to finding co-ordinates of ##B## and ##D##...
Anyway find my current approach on...
Summary:: Two parallel lines (same slope) - one intersects the y-axis, and the other doesn't.
Trying to find the intersection of either with a given geometric sequence.
The lines are:
y=mx
y=mx+1
The values on one or the other of the lines - but not both simultaneously - are to be completely...
A particle moves on a straight line. It starts at a point O on the line and returns to O 100 s later. The velocity of P is v m/s at time t s after leaving O where v= 0.000t^3- 0.015t^2 +0.5t.
1) Find the values of v at the times for which the acceleration of P is zero
I got when t= 21.1s, V=...
It's the second question.
Limiting equilibrium by a force of 5N means even friction is acting in the same direction. I don't understand how to calculate. Pls help
A woman on a sledge moves in a straight line across horizontal ice. Her initial velocity is 2 m/s. Throughout the journey her acceleration is given by a= -0.01t m/s^2, where t is the time from the start in seconds. Find the distance that she travels before coming to rest.
Iam getting the ans...
A goods train starts from rest at point A and moves along a straight track. The train moves with acceleration a m/s^2 at time t s, given by a=0.1t^2(6-t) for 0<t<6. It then moves at constant velocity for 6<t<156 before decelerating uniformly to stop at point B at t=165. Calculate the distance...
A particle starts at the origin and moves along the X- axis. The acceleration of the particle in the direction of the positive x-axis is a= 6t-c for some constant c. The particle is initially stationary and it is stationary again when it is at the point with x coordinate = -4. Find the value of...
In q(c) I calculated the distance using t= 4.25 S . I get S=-3.23×10^-3
I added 5.78m to this which is the distance between A and B, I get 5.78m. Is this the correct method to prove that the bird returns to A. Pls help
I don't understand q(e) . Am I not supposed to calculate the distance between the interval 0 to 2? The textbook ans only shows the interval between 0 and 22. Should I calculate the distance taming only the second equation of velocity?
A particle moves in a straight line. The velocity of the particle, v m/s, at time t s is given by v= -t^3+9t m/s for 0<t<5
a) Find the displacement of the particle from its original position, when t=5s
I got the ans for this by integration and limits 5 and 0 =- 43.8
b) work out the distance...
S for 0 to 2s = 20m
From 2s to 2.5s, I integrated v with limits 2.5 and 2 and got s=8.875m
So total distance would be 28.9m but the textbook ans is 29.9m. Iam not able to get 29.9 m
So recently I watched a video detailing how it is impossible to measure the speed of light in a straight line because it's not possible to synchronize two-time measuring devices without first knowing the speed of light.
But I was thinking if light can orbit a black hole in the photon sphere...
Let $P$ be a real polynomial of degree five. Assume that the graph of $P$ has three inflection points lying on a straight line. Calculate the ratios of the areas of the bounded regions between this line and the graph of the polynomial $P$.
This seems like a trivial question at first, but I am struggling how to get it right. If I have 2 GPS (lat, lon, height) observations P1 and P2 taken at some height from the Earth's surface, how do I calculate the straight line distance between them? I am using the picture below to illustrate...
Summary:: The set of values of ##b## for which the origin and the point ##(1, 1)## lie on the same side of the straight line ##a^2x+aby+1=0## ##\forall~a\in\mathbb{R},~b>0##.(a) ##a\geq1## or ##a\leq-3##
(b) ##a\in~(-3,~0)\cup(\frac13,~1)##
(c) ##a\in~(0,~1)##
(d) ##a\in~(-\infty,~0)##
I tried...
Let say we have two line ##a_1x+b_1y+c_1=0## and ##a_2x+b_2y+c_2=0##. Then pair of straight line equation is
##a_1a_2x^2+(a_1b_2+b_1a_2)xy+b_1b_2y^2+(a_1c_2+c_1a_2)x+(b_1c_2+c_1b_2)y+c_1c_2=0##
i.e ##ax^2+2hxy+by^2+2gx+2fy+c=0##
Now if we take ##a_1=0##, then the first line becomes...
Part B of the following problem seems to be fairly straightforward, but I can't seem to understand it properly. I might be overthinking the problem entirely.
Would anyone be willing to help?
I have a planetary system with planets orbitting a central star (circular orbits). I want to shoot a projectile P in a straight line from planet A to B and need to calculate the angle or vector to shoot the projectile P. I know the straight line of the projectile is physically not correct, but...
sz+tz*+r=0=say w
so w* = s*z* + t*z + r*=0
Now ,
w+w* = (s+t*)z + (t+s*)z* + r+r* = 0
= p*z + pz* + k = 0...eq(1) ( k is a constant or twice real part of w)
which is in complex straight line equation form i.e ab* + a*b + c = 0 ( a,b are complex number and c a real number.
Now, again...
Homework Statement
The diagram shows parallelogram ABCD. (you don't really need the diagram)
vector AB= (2 above, 7 below) and vector AC= (10 above, 11 below)
The point B has coordinates (5, 8)
(a) Work out the coordinates of the point C.
The point E has coordinates (63, 211)
(b) Use a...
Homework Statement
Train starts to travel in a straight line at increasing speeds. The first wagon passes next to me in 4 seconds. In how many seconds will the n-th wagon pass next to me?
Homework Equations
s= vo.t + 1/2at^2
v= vo + at
The Attempt at a Solution
First of all we didn't learn...
Homework Statement
Find the value of the parameter α for which the pencil of planes through the straight line AB has a common plane with the pencil of planes through the straight line CD, where A(1, 2α, α), B(3, 2, 1), C(−α, 0, α) and D(−1, 3, −3).
Homework Equations
Let Δ be a line given by...
In a graph , straight line intersects the parabola at(-3,9) & (1, 1) Then the equation is
A) x^2-2x+3=0
B) x^2+2x-3=0
C) x^2-3x+2=0
D) x^2-2x-3=0
I know that I can find the answer by substituting the known values to each options, but how to do it the proper way? We need at least three known...
Good day,
From the first attachment...
Why is "D" divided by "D1"? The book is saying that it is for normalizing "D" by the value of "D1".
It's not quite clear !
The whole formula is to find the slope of this graph.
Hey everyone,
I'm trying to plot a straight line for a bipolar junction transistor to find the room temperature, T, using my experimental results for the associated base-emitter voltage, ##{V}_{BE}## and collector current, ##I_C##. Here's the equation that I'm using:
$$ I_C = α_F {I}_{EO}...
Really basic question. I was talking with my friend and we started to get onto discussing lines when I said in three dimensions a straight line can be curved. He thinks of a straight line as a line with no curve, whilst I see it as the shortest possible distance from A to B, which in 3...
I was just thinking of basic definitions of geometry and i came to this question, so how could i prove that only one straight line passes through two distinct points.
Homework Statement
Let A = (1,2,5) and B = (0,1,0). Determine a point P of the line AB such that ||\vec{PB}|| = 3||\vec{PA}||.
Homework EquationsThe Attempt at a Solution
Initially, writing the line in parametric form\vec{AB} = B - A = (0-1,1-2,0-5) = (-1,-1,-5)\\
\\
\Rightarrow \vec{v} =...
Homework Statement
The problem is your typical N-body simulation, implemented using Python and Numpy. The implementation specifically calls for using the Euler-Cromer method. For this particular case I used the Sun and the first 4 planets of the solar system.
Essentially the problem is I'm...