Questions on the use of algebraic techniques to prove geometric theorems.

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Rotation of axes transformation as definition of vectors

Given a three-axes coordinate system ${1,2,3} $ by the right-hand rule, and a new coordinate system ${1',2',3'}$ , I know that one can define a vector $\vec{x}$ to be something that obeys the ...
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3answers
49 views

Calculus and analytic geometry question

Find the tangent of the angle in which the functions $x^3 $, and $x^2 $ intersect $(x≠0)$ . I find this question to be quite funny since the intersection point has two tangents going to it, with ...
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2answers
61 views

(I only need some hints)Find the vector equation of a space curve that represents an ellipse with the given center that lies in the given plane

Full disclosure, this is for a Calculus III graded homework set--though we are allowed to use any resources available to complete it. I feel I have a good understanding of space curves, though my ...
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3answers
25 views

solving for the left side for a double angle formula

$\cos t-\sin 2t=0$ Solve the left hand side so that it equals zero. Do I use $(2\sin t \cos {t})$ for $\sin 2t$?
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1answer
149 views

Does a square have an equation? [duplicate]

can you model a square in an equation ? like a circle for example $r^2 = x^2 + y^2$ and lets say we have a square with: centered at $(3,3)$ $2 \leq x \leq 4$ and $2 \leq y\leq 4$ can we somehow ...
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2answers
38 views

Finding the point that a normal line goes through

I have been stumped on a homework problem for quite some time and I'm hoping to get some help with it. The line from the origin to the point $(a, f(a))$ on the graph of $f(x) = \frac1{x^2}$ is ...
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1answer
29 views

How to Create a Plane Inside A Cube

I have a $e \times e \times e$ cube and I want to create random planes with equation $ax + by + cz + d = 0$ inside this cube. I will put random points on those randomly created planes as well. Here ...
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1answer
58 views

Proof for $\cos(\alpha)^2 + \cos(\beta)^2 + \cos(\gamma)^2 = 1$ in Euclidean space

What is the proof for this formula: $$ \cos(\alpha)^2 + \cos(\beta)^2 + \cos(\gamma)^2 = 1, $$ where $\alpha$, $\beta$ and $\gamma$ are the angles between a vector and the base of a right-handed ...
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1answer
20 views

Graph a line oriented by an specific angle?

I'm writing a software that plots gps data on a map, and so far it has been riddled with complex math problems, many of which I was able to fix by myself but this one I can't figure out. The software ...
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0answers
42 views

Cauchy-Schwarz on a Euclidian Space

I was thinking about this proof of the cauchy-schwarz inequality, I wanna show that $$|\langle u,v\rangle|\leq|u||v|$$. We know that, $$|\langle u,v\rangle| = ||u||v|\cos{\theta}|$$ where $\theta$ ...
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1answer
30 views

locus sections and circles--symmetry

A. Let L = {(x,y,z)|the distance from (x,y,z) to the y-axis is 6}. Describe what shape L is. So far, I have that it's a circle, but I'm not sure how to describe it fully. Would it be a circle that ...
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1answer
10 views

Finding the degree to cover a flare shape

I have a lamp shade frame that I want to cover with bamboo slats. The top is 12" around and the bottom is 33" around. The distance from the top to the bottom most part is 6". What degree, with minor ...
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1answer
43 views

Interesting Analytic Geometry Problem: Find internal angles given coordinates of final point and length of line segments

I have been mulling over a really interesting question in analytic geometry that is much harder than it first appears to be. Hope you can provide some insight into solving it. If you only know: ...
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1answer
70 views

Differential geometry question.

Please explain how to solve this question. Thank you:) And sorry for hand-writing.
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2answers
39 views

“Looping” equation

I'm looking for a equation that describes the shape of a "Looping" in the best way. I really don't know how to start here, as it isn't even a function (if it were, I could just use spline ...
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1answer
51 views

Rotate the Points on a Plane $P = ax+by+cz + d = 0$ parallel to $z = 0$ plane

I have a plane $P = ax+by+cz + d = 0$ and many points on that plane. I want to rotate $P$ so that it becomes parallel to $z = 0$ plane. Which method should I use? I know that the normal vector of my ...
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1answer
18 views

Find Points in The Cube

In the cube $ABCDA'B'C'D'$, we have $3\overrightarrow{AM}=\overrightarrow{MD}$ and $2\overrightarrow{D'N}=\overrightarrow{NB'}$. Find the points $M$ and $N$ in the cube; So, i can't find a way to ...
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0answers
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finding a locus in a $3$ dimensions

Given a Tetrahedron $OABC$ such that $O(0,0,0),A(a,0,0),B(0,b,0),C(0,0,c)$ ; $a,b,c$ are not zero. We build a plane $\pi$ that is parallel to $z$-axis and also to $AB$. Plane $\pi$ cuts plane $ABC$ ...
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2answers
177 views

The number of grid points near a circle.

There is a circle with center $(0, 0)$ and radius $r$. Let $n$ be the number of grid points inside or on the circle that at least one of its neighboring (up, down, left, right) grid points is outside ...
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1answer
84 views

How To Generate Random Points on the Positive Side of a Plane in 3-D

Edit: The question can also be interpreted as: How to generate random coplanar points in a cube? Here is what I am struggling with: I have a cube, whose origin is $(0,0,0)$ and one edge length ...
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3answers
74 views

Trigonometry, knowing 3 sides how to find the height?

I have a mathematician problem where, I knew the 3 sides of a triangle, with these sides I can figer out what type of type of triangle is. What I realy want to find is the height of the triangle and ...
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1answer
17 views

Concept of parallelism in analytic terms

Below I cited a passage from Apostol's Calculus. I don't understand how to use the identity to show that two lines with equal slopes are parallel. Concepts such as perpendicularity and parallelism ...
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2answers
40 views

This question on right triangles. [closed]

a ladder 10 ft. in length is leaning against a wall 8 ft. above the ground. a 6ft. guy tries pass underneath the ladder 1 ft. away from the wall. will he be able to pass underneath the ladder.
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How will I get the point of intersection?

I'm confuse on how will I get the point of intersection of these two equations: $x^2+y^2+5x+y-26=0$ and $x^2+y^2+2x-y-15=0$ I tried using the elimination method but I can't get it.
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2answers
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find the equation of the locus of a moving point which is always equidistant from the y-axis and the point (-6,4)

Do you know how to solve its equation? Already solved some locus problems that gives points but not in the y or x axis problems.
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1answer
26 views

Formula to plot a non-linear graph

Firstly, thank you very much in advance. I need to express a non-linear graph comprised (piecewise) of the following linear elements: A line from $(x=0,y=100)$ to $(x=10,y=100.5)$ A line from ...
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1answer
25 views

A rectangle $OACB$ with two axes as two sides,the origin $O$ as a vertex is drawn in which the length $OA$ is four times the width $OB$…

A rectangle $OACB$ with two axes as two sides,the origin $O$ as a vertex is drawn in which the length $OA$ is four times the width $OB$.A circle is drawn passing through the points BC and touching ...
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2answers
84 views

Equation of the Circle

How to find the equation of a circle if the givens are the: Case 1: Tangent to $2x + 3y + 13 = 0$ and $2x - 3y - 1 = 0$; contains $(0,4)$ Case 2: Tangent to $x - 3y - 7 = 0$ and $3x + y - 21 = 0$; ...
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1answer
28 views

analytic geometry , Orthogonal projection

II) take the point P = (2,1,0) , the line r : X = (0,0,0) + t(2,1,0) and the plane B: x + y + z -3 = 0 . For each point Q let it be Q' his orthogonal projection to B .Find the Q points that outputs ...
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2answers
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analytic geometry , perpendicular planes and a line

Find a equation to the plane that contain the line $X = (1,0,2) + t(4,1,0)$ and is perpendicular to the plane $A : 3x + y + z = 0$
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2 line segments of similar lengths cut off by logarithmic functions

We have the function $f(x) = \log_{1/3}(x+3), g(x) = 2 - \log_{1/3}(x), h(x) = -3+\log_{1/3}(x-1)$. The line $y=p$ with $-1 < p < 0$ is divided into 2 line segments with equal length by ...
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1answer
49 views

triangle proof: intersection of $w_\alpha$ and $m_a$ is outside $\Delta ABC$

I try to proove, that the intersection $M$ of $w_\alpha$ (angle bisector) and $m_a$ (perpendicular bisectors of $a$) of any triangle $\Delta ABC$ is always outside $\Delta ABC$, except $\Delta ABC$ is ...
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111 views

Computing Euler Angles from Direction Cosines Vector

My problem simply as the following: Suppose we measured the orientation of a plane object with respect to a reference fame. (where the reference frame parallel to plane frame). The unit normal vector ...
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Quadric surfaces classification

I have an exam soon on quadrics. However, I don't have enough exercises, especially ones involving finding standard form of quadrics using affine and orthogonal transformations - rotation, translation ...
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1answer
33 views

Find the equation of a plane

Find the equation of a plane that passes through point $P(1,5,1)$, and is perpendicular to the planes $2x+y-2z=2$ and $x+3z=4$ My only guess so far is that we can obtain the plane's normal vector ...
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3answers
70 views

Parametrization of $y^2 - x^2=1$

I have found parametrizations for the level curve $y^2-x^2=1$, however, I have a question regarding one of them. From the Pythagorean trigonometric identity $\cos^2 x + \sin^2 x =1$ we obtain ...
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0answers
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Equations for Intersection of Plane and circle

I am having a problem in getting the related equations for an intersection between a point on a plane and the edge of a circle in 3D space. Any suggestions? Or also a tangent plane and a point on a ...
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3answers
42 views

Analytic Geometry question, planes and lines

Let there be a plane going through three points $(0,2,-9), (0,-1,0), (-\frac{3}{m},1,-3)$. For which value of $m$ is the line $l: (3,0,-9)+t(2m,-5,7)$ onto (or 'inside') the plane? Not sure how to ...
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2answers
110 views

How to find vertex of a parabola from its second degree equation

Given a parabola with second degree equation as $$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 $$ assume that this isn't degenerate case, and $B^2-4AC=0$ How can I find its vertex position?
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2answers
64 views

Equation of a circle given one point and two lines

Find the equation of the circle that pass through $(2,3)$ and are tangent to both the lines $3x - 4y = -1$ and $4x + 3y = 7$.
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2answers
58 views

Finding point on ellipse equally distant from two other points on the ellipse

I have an ellipse with two points on it: A and C (with known coordinates). Point O is the center of the ellipse (coordinates are given). I need to find coordinates of point B which also lies on the ...
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Volume of a regular tetrahedron vs Volume of a sphere

I have the following question: given a regular tetrahedron and a sphere that goes through the middle of all sides of the regular tetrahedron, which has a bigger volume? and what is the ratio of ...
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Circle touching the y-axis passing through two points

How to find the equation of the circle touching the y-axis and passing through two points?
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1answer
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Plane equation question!

Could anyone explain me how to do tasks like this one: Plane is intersecting Oy axis when $y = 3$ and line equation is $ 2x + 4= y-2=z$ belonds to plane. Write plane equation.
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A sphere packing problem

Suppose there is a large sphere of radius $R$. We want to pack it with smaller spheres. The volume of the smaller spheres change depending on where they are situated in the larger sphere. A smaller ...
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1answer
100 views

A moving point has its distance from (1,3) always one-third of its distance from (8,2). Find the equation of its Locus.

A moving point has its distance from (1,3) always one-third of its distance from (8,2). Find the equation of its Locus. My equation displays a circle formed by the loci, I don't know if it's right. ...
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2answers
41 views

**Each Pair Bisects the angle between the other pair** then $pq=?$

If the pair of straight line $x^2-2pxy-y^2=0,x^2-2qxy-y^2=0$ are $\ni$ Each Pair Bisects the angle between the other pair then $pq=?$ I do not understand geometrically,mathematically the boldly ...
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1answer
55 views

Analytic geometry textbook introduction.

I need the help of others concerning a good-rigorous analytic geometry textbook. (high school level.) Thank you for the help!
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A question on co-ordinates of intersecting lines…Given in picture below

Please do also MENTION how you got the solution.........
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Analytic geometry simple question

need help in this. In a right triangle in the three-deminsion plane $ABC$, A=$(2,-3,4)$, B=$(1,-1,5)$. Find $C$ if its known $C$ is on the line $L: (1,5,-2)+t(3,0,-2)$. What I did was finding the ...