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

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1answer
90 views

show that the straight lines $(a^2-3b^2)x^2+8abxy+(b^2-3a^2)y^2=0$ form with the lie $ax+by+c=o$ an equilateral triangle

show that the straight lines $(a^2-3b^2)x^2+8abxy+(b^2-3a^2)y^2=0$ form with the lie ax+by+c=o an equilateral triangle whose area is $\frac{c^2}{\sqrt{3}(a^2+b^2)}$ is there any other way to solve ...
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2answers
256 views

equation representing 2 straight lines

Let us assume this equation is given to us we have to factorize it $$12x^2 +7xy-10y^2+13x+45y-3=0$$ By solving we get that this represents two straight lines. But how to factorize it? Is there a ...
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0answers
105 views

Linear separability in a 2×2×2 cube

Essentially, linearly separable points are just those corners that can be cut off with just one slice as marked out by a hyperplane. Can someone please explain how many cases there are for a ...
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3answers
511 views

Why do we believe the equation $ax+by+c=0$ represents a line?

I'm going for quite a weird question here. As we know, the equation in Cartesian coordinates for a line in 2-dimensional Euclidean geometry is of the form $ax+by+c=0$. I'm wondering why do we ...
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3answers
955 views

Given middlepoints of a sides of triangle, find vertices

If $(2, 1)$, $(3, 3)$ and $(6, 2)$ are the middlepoints of a sides of a triangle, what are the coordinates of a vertices of a triangle? This part of the book deals with midpoints, with formula: ...
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1answer
479 views

How to show that a line pass through a point?

How to show that a line pass through a point? Two fixed straight line $OX$ and $OY$ are cut by a variable line at the points $A$ and $B$ respectively and $P$ and $Q$ are the feet of the ...
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1answer
52 views

If $l(x,y)=0$ and $l'(x,y)=0$ intersects in $P_0(x_0,y_0)$ then $ \lambda l (x,y) + \lambda ' l'(x,y)=0 , \quad ( \lambda +\lambda ' =1 )$

I'm reading "What is Mathematics?" and found this question. let $l(x,y)=0$ represents the equation $ax+by+c=0$ of a line $l$,and so does $l'(x,y)=0$ . now let $\lambda + \lambda ' =1 $. Show that, if ...
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4answers
197 views

How do I find the center and radius of this circle? [closed]

How do I find the center and radius of this circle? $$4x^2+4y^2+24x-16y+41=0$$
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3answers
228 views

Define “y” value from the equation of circle

Let's take a circle. It has the following general equation to describe it: $(x-u)^2+(y-v)^2=r^2$ ,where $u,v$ is the coordinates of the center of the circle, and $r$ is the radius of the circle. If ...
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2answers
125 views

how do i calculate coordinates of point on rectangle based on angle

If i want to find out what is x and y of point lies of a line drawn from center of the rectangle to outward at certain angle. take a look at picture below.
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1answer
1k views

Ellipse in Cartesian and in Polar Coordinates

So I was studying about ellipses in Polar Coordinates, and the book said Let F be a fixed point, and l be a fixed line in a plane. Let e be a fixed positive number. The set of all points P in ...
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1answer
168 views

Find the locus of the point which divides segment AB internally in the ratio 1:2.

A and B are variable points on X and Y axis respectively,such that l(ab)=4.Find the locus of the point which divides segment AB internally in the ratio 1:2. I think that it must be a circle . or ...
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1answer
114 views

Problem regarding lines (Analytic Geometry)

Here is the problem: The base of a triangle has a fixed position and its length is constant and measures a. The difference of the squares of the other two sides is constant and measures $b^2$. ...
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2answers
1k views

How to prove the parallelogram law by using geometric method?

How to prove th parallelogram law by using geometric method? NOT ALGEBRA
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3answers
2k views

How to find a perpendicular vector

After thinking longing i can't figure it out no matter what.. So i have 3d line starts (0,0,0) and ends (3.5,3.5,2.5) so therefore has length of about 5. Now how do i find out vector that is ...
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1answer
124 views

Direction cosines

I have two vectors, $\vec A$ and $\vec B$ that meet at point R. The vector $\vec {RA}$ has a magnitude 2km and direction cosines $\cos(\alpha)$=0.768, $\cos(\beta)$=0.384, $\cos(\gamma)$=0.512. The ...
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1answer
4k views

Proof of vector addition formula

Two vectors of lengths $a$ and $b$ make an angle $\theta$ with each other when placed tail to tail. Show that the magnitude of their resultant is : $$r = \sqrt{ a^2 + b^2 +2ab\cos(\theta)}.$$ I ...
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1answer
150 views

Find the slope or angle to the horizontal of an internal common tangent of two circles

I'm looking for a general (hopefully computationally efficient) algorithm for the problem in the title, given the centers and radii of the circles in question. If it matters, I am always looking for ...
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1answer
3k views

Given slope of the line, find inclination

What is the inclination of the line joining $(3, 0)$ and $(2, \sqrt{3}) $ Answer: $\frac{2\pi}{3}$ $m = tan(\alpha)$ So, $$m = \frac{y_1 - y_2}{x_1 - x_2}$$ $$m = \frac{0 - \sqrt{3}}{ 3 - 2} = ...
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1answer
27 views

Can lines be defined with its slope and a point on it definitely?

If I get the slope of a line, and one point that is on it, then, are they define exactly ONE line? The point–slope form of linear equations ($y - y_1 = m( x - x_1 )$) need only the slope and the ...
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1answer
112 views

line parallel to plane, but not on plane.

I need to find a plane that goes through the points $A=(2,0,2)$ and $B=(4,1,0)$, that is parallel to the line? $$r(t) = (0,3,-2) + t\langle1,-1,1\rangle$$ or if you want it in parametric equations: ...
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1answer
423 views

Intersection of two planes and another plane parallel to the intersection.

I have two questions: $1)$ How can I find the line of intersection between the planes $$x+ 2y +z =4 \\ \mathrm{and} \\ 2x+y-z=5$$ $2)$ How do I find an equation for a line that goes through $A = ...
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1answer
135 views

Are cartesian coordinates “more fundamental” than other coordinates, and are they inherently tied to $\mathbb R^n$?

Are the Cartesian coordinates more "fundamental" than other coordinate systems? When someone says $\mathbb R^n$ do we implicitly mean the set of points PLUS Cartesian coordinate system? Sometimes I ...
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0answers
196 views

Formula for intersection of “power” curve and parabola.

EDIT I have edited this question to make it more clear. I have spent quite some time trying to find this on Google, but haven't succeeded. I need the formula(s) to determine the intersection ...
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4answers
2k views

How do I move through an arc between two specific points?

I've found many answers to similar questions here, but I'm still stuck. I want to move an object from point sx,sy to point dx,dy through an arc that bulges by distance b from the line straight ...
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4answers
2k views

Find the equation of a parabola (in general form)

Find the equation of the parabola with axis parallel to the $y$-axis, passing through $(1/2,-5/2),(3/2,-9/4)$ and $(-7/2,3/2)$.
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2answers
518 views

Why is the locus of the centres of the circles passing through two points is the perpendicular bisector of the two points?

Why is the locus of the centres of the circles passing through two points is the perpendicular bisector of the two points?
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1answer
73 views

Let $p_1,\dots, p_5$ be five points, no three of which are collinear.

1) Let $p_1, \dots, p_5$ be five points, no three of which are collinear. How many lines contain two of these five points? 2) If no four of the five points are coplanar, how many planes contain three ...
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2answers
1k views

How to calculate the intersection of two planes?

How to calculate the intersection of two planes ? These are the planes and the result is gonna be a line in $\Bbb R^3$: $x + 2y + z - 1 = 0$ $2x + 3y - 2z + 2 = 0$
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1answer
390 views

How to find the angle between vectors which is not necessarily the smallest angle

I understand that in order to calculate the angle between two vectors one does the arccos of the results of the dot product divided by the product of the magnitude of the two vectors. However, this ...
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1answer
1k views

Angle between line and a plane

I want to calculate the angle between the plane with a normal $N = [N_x,N_y,N_z]$ and the vector $V = [V_x,V_y,V_z]$ and I used this formula for angle $$\alpha = \arccos \frac{V \circ N} {|V|\; ...
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1answer
541 views

find the rate of change of the area of triangle pulled by three people from its sides

this is the problem of my curious mind(I am it's designer!) . three people each having the rope attached by the end of the 3 sides of triangle ABC , pull the triangle with speed U in the direction ...
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4answers
44 views

compute perpedicular vector component

I have two vectors $a=(a_x, a_y)$, $b=(b_x, b_y)$ . How can I compute the component vector $p$ of vector $a$ perpendicular to vector $b$? (how to compute vector $p$)
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3answers
125 views

Help with Geometry (sphere) question

Consider a sphere with the following equation: $$(x - 9)^2 + (y + 5)^2 + (z - 2)^2 = 49$$ answer all the questions below a. What is its center? b. What is its radius? c. True or false. (3, –3, 5) ...
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1answer
85 views

Is a flat morphism of complex algebraic varieties open in the analytic topology?

Let $X,Y$ be algebraic varieties over $\mathbb{C}$. A morphism $f:X\to Y$ induces a morphism $f^{an}:X^{an}\to Y^{an}$ between the associated complex analytic spaces (actually, I am interested only in ...
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2answers
89 views

How to find $P_1$ in $(x,y)$ form

From following diagram, $A_1$ is center of circle of radius $r$. All distances are in coordinate system $(x,y)$. Distance from $A_1P_2$ is known. Distance $A_1,A_2,A_3$ is also known from origin. I ...
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2answers
178 views

Find the equation of a line

Find the equation of the line through (12/5 , 1), forming with the axes a triangle area of 5. There are 4 solutions and how can i get it?
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2answers
260 views

Why is a projection matrix symmetric?

I am looking for an intuitive reason for a projection matrix of an orthogonal projection to be symmetric. The algebraic proof is straightforward yet somewhat unsatisfactory. Take for example another ...
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1answer
85 views

Is the result of the actions $\left((\vec A+\vec B) \times (\vec A\times \vec B)\right)\cdot(\vec A \times \vec B)$ depends by $\vec A$ and $\vec B$

I want to show that this action not depend by A and B vectors, I know that cross product of the same vector by itself is $0$. $$\left((\vec A+\vec B) \times (\vec A\times \vec B)\right)\cdot(\vec A ...
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1answer
3k views

Find plane which parallel to two vectors $L_{1} ( 3,1,10)$ and $L_{2}(1,-1,1)$ passes through a point $M(7,-10,3)$

I`m trying to find a plane which parallel to two vectors $L_{1} ( 3,1,10)$ and $L_{2}(1,-1,1)$ passes through a point $M(7,-10,3)$ what I tried to do is to create $L_{1}L_{2}$ vector then to create ...
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2answers
901 views

Find the projection of the point on the plane

I want to find the projection of the point $M(10,-12,12)$ on the plane $2x-3y+4z-17=0$. The normal of the plane is $N(2,-3,4)$. Do I need to use Gram–Schmidt process? If yes, is this the right ...
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2answers
771 views

Find the point of intersection of the straight line $\frac{X+1}{4}=\frac{Y-2}{-2}=\frac{Z+6}{7}$ and plane $3X+8Y-9Z=0$

Find the point of intersection of the straight line $$\frac{X+1}{4}=\frac{Y-2}{-2}=\frac{Z+6}{7}$$ and plane $3X+8Y-9Z=0$ the point of the line is $M(-1,2,-6)$ and direction vector of the line is ...
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1answer
519 views

Explain what are the projections of the point $P(a,b,c)$ on the planes of the coordinate system.

I want to explain what are the projections of the point $P(a,b,c)$ on the planes of the coordinate system. the meaning is that if it on $XY$ plane so it will be $P(a,b,0)$ and so on? Thanks!
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2answers
962 views

Check whether the three vectors $A(2,-1,2),B(1,2,-3),C(3,-4,7) $ are in the same plane

I want to check if three vectors are in the same plane, the vectors being $$A(2,-1,2),B(1,2,-3),C(3,-4,7). $$ What I did so far is to create vector $AB ( -1,3,-5)$ and build the plane equation ...
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2answers
409 views

Angle between planes

If the angle between two planes is $\alpha$ , why is the angle between normal of the two planes is $\pi - \alpha$ ? Also Why angle between a line and normal to a plane is $\pi/2 -\alpha$ if angle ...
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1answer
58 views

Conic section - hyperbolic path

I got that equation of path is conic section $u=\frac{1}{3c}(1+2\cos\theta)$ where $c$ is constant and one vertex of hyperbola is $(-c,0)$ and $u=r^{-1}$. So, $r=\frac{3c}{1+2\cos\theta}$. Since ...
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4answers
413 views

Some theorems in euclidean geometry have incomplete proofs

I have seen that, in euclidean geometry, proofs of some theorems use one instance of the 'geometric shape'(on which the theorem is based) to proof the theorem. Like, the proof of 'A straight line ...
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1answer
227 views

3D Road - Rotate around 3d curve

First of all, I'm not sure whether to post this on stackoverflow or here, but since there's some mathematics needed here (especially at the end of this question) I posted it here. I'm given a ...
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2answers
70 views

Calculate $\bigtriangleup$ ABC where $A(-2,-3,0)$,$B(-1,0,5)$,$C(4,2,2)$

I want to calculate $\bigtriangleup$ ABC where $A(-2,-3,0)$,$B(-1,0,5)$,$C(4,2,2)$ What I did was to mark the triangle vertices randomly 1) calculate the middle of AB ( I call it G ) to find the ...
2
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1answer
34 views

Find the vector by the following criteria

I want to find the vector that meets the following: $$X\parallel (2,1,-1)$$ $$X*(2,1,-1)=3$$ what I did so far is : $$2x+y+z=3$$ I know that parallel vectors the angle is $180$ or $0$. how to ...