What is Analytic geometry: Definition and 49 Discussions
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
Analytic geometry is used in physics and engineering, and also in aviation, rocketry, space science, and spaceflight. It is the foundation of most modern fields of geometry, including algebraic, differential, discrete and computational geometry.
Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes three dimensions. Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space (three dimensions). As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometrical shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations. That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom.
In my recent study of Conic Sections, I have come across several proofs (many of those comprise Dandelin spheres) showing that the cross-section of a cone indeed follows the focus-directrix property:
"For a section of a cone, the distance from a fixed point (the focus) is proportional to the...
Currently, as far as I know, the two main ways to express any given point on a plane is through either cartesian plane or polar coordinates. Both of which requires an ordered pair of two numbers to express a point. However, I wonder if there exists such a system that could express any given...
I'm trying to design a simple mechanism in CAD and got stuck on this problem.
Consider a lever (blue shape) that pivots at one end (blue circle). The other end rests on a piston (pink rectangle). In my application the lever is pushing the piston (the lever pushes through the full travel of a...
Consider the following scenario:
Given that points ##M## and ##N## are the midpoints of their respective line segments, what would be the fastest way to determine what percentage of the squares total area is shaded purple?
I managed to determine that the purple shaded area is ##5\text{%}##...
I started off by indicating ##p=-\frac{1}{m}## since it's perpendicular. The sum ##m+p## is now ##\frac{m^2-1}{m}##.
Honestly, I can't go beyond that. The interceptions with the y-axis are of course unuseful, I tried algebraically intersecting the two lines but I came up with nothing... and I...
I was just thinking, if is said to me demonstrate any geometry statement, can i open the vector in its vector's coordinates? I will say more about:
For example, if is said to me: Proof the square's diagonals are orthogonal, how plausible is a proof like?:
d1 = Diagonal one = (a,b,c)
d2 =...
Every explanation about scaling a 2D vector, or equivalently having a line segment PQ on cartesian plane and then find a point R on the line PQ satisfying PR/PQ = r (fixed given r) starts with that one specific case in the picture. A formula for the coordinates of R is then given for that case...
1.To shift the graph of a function :
Vertical Shifts : ## y=f(x) +h## where the graph shifts ##k## units up if ##k## is positive and downwards when ##k## is negative.
Horizontal Shifts : ##y=f(x+h)## where the graph shifts to the left by ##h## units when positive and to the right when ##h## is...
My book states the following :
##{\frac{(y_1)-(y_2)}{(x_1)-(x_2)}}*\frac{(3+1)}{(0-0)} = -1## ...(1)
##\implies y_1 - y_2 = 0##
(A) Is this a valid deduction?
Context :
The Problem :If ##A(0,-1)## and ##B(0,3)## are two opposite vertices of a square, then find the other two vertices...
Homework Statement :[/B]
The following expression stands for the two angular bisectors for two lines :\frac{a_{1}x+b_{1}y+c_{1}}{\sqrt{a_{1}^{2}+b_{1}^{2}}}=\pm \frac{a_{2}x+b_{2}y+c_{2}}{\sqrt{a_{2}^{2}+b_{2}^{2}}}\qquad
Homework Equations
The equations for the two lines are :
##a_1x + b_1y +...
Given the coordinates ##P = (3,4)## , find the coordinates of ##P"(x',y')## when the origin is shifted to (1, –2), and the axes are rotated by 90° in the clockwise direction.
I attempted to solve this problem using the following formulas :
##x = X + h## and ##y = Y + k## for translation of the...
My question is not a math question.
I know about the calculus sequence (CAL 1, 2 and 3). I plan to go through all 3 in time. There is no rush for me. However, I know there is a course by the title of Calculus and Analytic Geometry. I want to know when this course is given. Is it given after...
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} =...
In a triangle ABC: A(-2,7) and C(7,-5).
The length of the altitude of AC is 5, and the length of the altitude of BC is the square root of 45. I wish to find the vertex B, given that it is below the line AC.
I need your help, I have no idea how to approach this.
Thank you in advance.
I have...
Homework Statement
Let S be a sphere with the equation ##(x-2)^2+y^2+z^2=2 ## and let p a line which satisfies the condition ## p \in (\Pi \cap \Sigma) ## where ##\Pi## and ##\Sigma## are planes with equations:
##\Pi :x+z=2##
##\Sigma: 5x-2z=3##
a) Show that S and p have exactly one common...
I am actually very confused on how to solve the problem. Do I just find the distance between the lines? How do I incorporate the 500 Newtons into the problem? Really confused.
Hi. I'm trying to proof the image formation property of a concave spherical mirror. I know you can do this easily with a particular choice of rays (namely one that hits the vertex and one that passes through the center of the sphere) but I would like to show that a generic ray yields the same...
Homework Statement
Find a curve that passes through point A(2,0) such that the triangle which is defined with a tangent at arbitrary point M, axis Oy and secant \overline{OM} is isosceles. \overline{OM} is the base side of a triangle.
2. The attempt at a solution
Function passes through point...
Homework Statement
Find area bounded by parabola y^2=2px,p\in\mathbb R and normal to parabola that closes an angle \alpha=\frac{3\pi}{4} with the positive Ox axis.
Homework Equations
-Area
-Integration
-Analytic geometry
The Attempt at a Solution
For p>0 we can find the normal to parabola...
The following passage has been extracted from the John Stewart's English translated version of the "Sir Issac Newton's two Treatises: Of the Quadrature of Curves, and Analysis by equations of an infinite number of terms" http://archive.org/details/sirisaacNewtons00stewgoog:
Here Newton...
I'm planning to write about analytical geometry, but I'm unsure about where to start. The course that I've taken begins with the definition of a vector, it then proceeds to develops all the vector algebra and all properties. After it reaches the cross product and triple product it begins to...
Hello, I am studying for an analytic geometry final but I am totally lost for this problem... We didn't even cover this topic in class (my prof didn't show up for class for two weeks) and I have no clue on how to do it. If anyone could help I would appreciate it.
Question: Prove that the...
Hello all, I was accepted into a program that allows me to take a free college course and I was hoping to take a math/science course. I am most interested in Physics, but I don't know Calc so that isn't really an option, and I was thinking of taking Calculus and Analytic Geometry. However, I am...
So I am trying to test into the Calculus with Analytic Geometry class at my school. I've already taken the first AP Calculus class my senior year of high school (but wasn't particularly concerned with my grade, and never took the test, since it was an extra curricular class for me). Then I took...
Hey so I have a math test tomorrow and I'm really good in math, I just need some small clarification, so here's my problem.
V(-2,k) P(4,9) slope of 2/3
so I decided to find y=mx+b with P(4,9) because after I find B I could do the same with V(-2,k)
so I did the equation and B came up 7...
I am always entranced when I read in calculus books about various curves like lemniscates, cardioids, the spiral of Archimedes, the witch of Agnesi, and similar things. But in calculus books they are generally just little asides or relegated to the exercises. Presumably there is more to analytic...
Hello PF! After months of eye shopping, I couldn't help but join this awesome community myself. I am a high school senior entering college soon, who is very excited to take college-level physics! I've done some research of what classes I should take, and came upon a conflict of scheduling...
The curve in the figures above is the parabola $y=x^2$. Let us define a normal line as a line whose first quadrant intersection with the parabola is perpendicular to the parabola. Five normal lines are shown in the figures above.
For a while, the $x$-coordinate of the second quadrant...
Question:
Find the point in the plane x+2y-3z=1 with minimum distance to point (1,1,1).
Attempt at resolution:
Well, techinically, I already have the means to solve this.
I can find the distance between the point and the plane to be 1/sqrt(14) and then I can either solve for the sphere...
Question:
What are the vector, parametric and symmetric equations of the angle bissector of angle ∠ABC, given that A=(1,2,3), B=(3,4,5) and C=(6,7,0).
Attempt at resolution:
Well, I defined some D=(d_1,d_2,d_3) to be a point in the angle bissector, and two lines r:X=(3+2a,4+2a,5+2a) and...
Hello,
Analytic geometry has provided us with such profound tools for thinking that it is hard to imagine what thinking must have been like before we had such tools. Two particular developers of these tools are Pierre de Fermat and Renee Descartes in 17th century France.
I would like to...
I'm trying to help a friend but I don't remember any of this, so it'd help us both. Book recommendations are also welcomed.
Given the vector equations $$r: \begin{cases} x = 2 - \lambda, \\ y = 1 + 3 \lambda, \\ z = 1 + \lambda, \end{cases} \text{ and } s: \begin{cases} x = 1+t, \\ y = 3+4t, \\...
Homework Statement
Find the equation of the circle whose centre lies on the x-axis and which passes through points A (6,0) and B (0,10).
Homework Equations
The Attempt at a Solution
I drew a diagram of the circle and determined that the line AB has gradient 5/3. Its perpendicular bisector...
Homework Statement
.
1. a hall that is 10 ft. wide has a ceiling that is a semi-ellipse. the ceiling is 10 ft high at the sides and 12 ft high in the center find its equation with the x-axis horizontal and the origin at the center of the ellipse.
Homework Equations
x^2/a^2 + y^2/b^2 = 1...
Hello! I'm looking for a good analytic geometry textbook. I'm studying circumferences and sphere and I'm using the book Analytic Geometry: A vector approach from Wexler but I find this book quite bad. I want some suggestions!
Thank you.
Homework Statement
Points S and T lie on the line 2x+y=3. If the length ST is 13 units, and the coordinates of S are (2, -1), determine all possible coordinate pairs for T, correct to two decimal places.
For simplicity, determine the coordinates for T, relative to S from the left.Homework...
Okay so I am reading a book on analytic geometry because my algebra class is starting to really bore me, and I ran across this:
slope = m = -1/2 (easy enouph)
but then it states "since the slope is the tangent of the angle of inclination θ, we have: tan θ = -1/2
okay so I guess my...
I am a sophomore at Penn State and I am taking Calculus for the second time, I didn't fail the first time, but I didn't want to take the chance I might get a D so I dropped it. Anyhow I am looking a great Math book to learn from while I am learning it again in class. We are using Calculus: Early...
Find the vertices, foci, and eccentricity of the ellipse.
1/2x^2 + 1/8y^2 = 1/4
i'm a little confused on how to clear the fraction and set the equation up to make it equal 1.
I tried to divide everything by 4 to clear it but 8 can't divide into 4 and if I use a higher number than 4 the...
Hello.
I am a freshmen student in mathematics at Moscow State University (in Russia) and I'm confused with placing the subject called "analytic geometry" into the system of mathematical knowledge (if you will).
We had an analytic geometry course in fall; now we are having a course in linear...
Homework Statement
In the figure, the orange line and the green line are perpendicular and intersect on the x-axis.Find the coordinates of P.
http://tinypic.com/r/2wn7m2v/3 go here for the image please.
Homework Equations
slope-point formula, m1*m2=-1 FOR PERPENDICULAR LINES
The...
I'm struggling to find the method to use when trying to get the normal vector n to a (flat) surface at a specific point. Every textbook I've researched has the same scenerio, where they show how to get a plane given a point and normal vector, but I have the opposite problem, where I know the...
Homework Statement
http://img196.imageshack.us/img196/5391/29069034.jpg
Homework Equations
only analytical methods
The Attempt at a Solution
I found k to be =2
I don't have any idea how to calculate angle A.
Please need some help.
Thank You.
Hi all! I've appreciated (and lurked :-) this forum for a while. While waiting to find a question I'm able to answer, it's time for my first post (hopefully the first of many).
Here's the problem: I'm trying to teach myself euclidean geometry and a found a tough (for me) question to answer in...
Homework Statement
Find the centre and radius of the circle x²+y²-4x+2y+6=0
I have the solution. The circle is no defined because r² = -1 is impossible.
But... how do I even DO that equation to get the answer -1?!
Hi all. I have a analytic geometry question that I need a bit of help with.
consider the concentric circles with the equations:
x^2 + y^2 = 9
and
x^2 +y^2 = 4
A radius from the center O intersects the inner circle at P and the outer circle at Q. The line parallel to the x-axis through P...