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

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0
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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 ...
0
votes
1answer
107 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: ...
0
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1answer
409 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 = ...
3
votes
1answer
134 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 ...
1
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0answers
193 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 ...
1
vote
4answers
1k 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 ...
1
vote
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
485 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?
0
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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 ...
1
vote
2answers
966 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
378 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 ...
1
vote
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|\; ...
4
votes
1answer
524 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 ...
0
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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
123 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) ...
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 ...
2
votes
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 ...
1
vote
2answers
172 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?
5
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2answers
249 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 ...
3
votes
1answer
84 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 ...
0
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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 ...
2
votes
2answers
895 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 ...
2
votes
2answers
647 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 ...
0
votes
1answer
509 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!
2
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2answers
931 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 ...
0
votes
2answers
407 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 ...
0
votes
1answer
57 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 ...
6
votes
4answers
399 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 ...
0
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1answer
226 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 ...
0
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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
votes
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 ...
3
votes
1answer
174 views

Parabolas intersecting in integer points

Can you construct an example of two different parabolas (with integer coefficients) that intersect at three integer points? An integer point is a point $(x,y)$ where both $x$ and $y$ are integers.
0
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3answers
2k views

Find the equation of the line thru a point forming with axes a triangle given an area (Read the question on the desc)

Find the equation of the line thru (2,2) and forming with the axes a triangle of area 9.
0
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2answers
68 views

Find the vector that meets the following criteria

I want to find the vector $X$ by the following lines: $$(1,-3,5) \cdot X=49$$ $$(4,1,-1) \cdot X = 0$$ $$(2,0,-3)\cdot X=-9$$ I would like to get some advice how to find him. Thanks!
0
votes
1answer
40 views

For which values ​​of M vectors$A(m-4,2,2m-12),B(2,m-12,2)$ are orthogonal

I want to find for which values ​​of M vectors$A(m-4,2,2m-12),B(2,m-12,2)$ are orthogonal. what I did is to do $A*B=0$ and the result was $m=7$ then I inserted $7$ and tried to check if they are ...
0
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1answer
99 views

Proving $AB$, $AC$ perpendicular to each other when vectors are median

I want to prove this claim: Triangle $ABC$ with $A(2,4,6),B(6,2,2),C(0,0,0)$ median, $AC$ and $BC$ perpendicular to each other. What I did is to the $AB$, $BC$ make a dot product and thought it ...
1
vote
1answer
2k views

Proving a triangle is a right triangle given vertices, using vector dot product

I want to to show that this triangle is a right triangle. I know that the dot of the vectors need to be $0.$ I tried to dot between them but I don't get zero. Claim: Triangle $\bigtriangleup ...
0
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2answers
46 views

Will the ball hit the wall?

There is a ball starting at point $A$ going forward in the direction towards point $B$ (so it moves along the $(AB)$ line). A wall is represented by its two ends $W_1$ and $W_2$. I have to solve in ...
1
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4answers
82 views

A proof in circles

The question is this: Show that the equation of a straight line meeting the circle $ x^2 + y^2 = a^2 $ in two points at equal distances $d$ from a point $(m, n)$ on its circumference is $ mx + ny - ...
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5answers
355 views

Geometric visualization of covector?

How could I geometrically visualize a linear functional?
3
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1answer
225 views

Given pairwise distances between points in space how can the dimension of the space the points lie in be determined if the data is incomplete?

My problem concerns a finite set of points in a space, for example (A,B,C,D,E). The information I have on the points is the distances of separations between the points and it comes in two sets. The ...
2
votes
3answers
112 views

How prove this inequality for $a,b,c,d$ are real numbers

let $a,b,c,d$ are real numbers,show that $$2\sqrt{a^2+c^2}+\sqrt{a^2+c^2+3(b^2+d^2)-2\sqrt{3}(ab+cd)}+\sqrt{a^2+c^2+3(b^2+d^2)+2\sqrt{3}(ab+cd)}\ge6\sqrt{|ad-bc|}$$ This problem is creat by China's ...
1
vote
2answers
55 views

Find equation of plane formed by a point and line

It is required to find the equation of a plane $Q$ formed by point $B\,(5,2,0)$ and the line (d) of parametric equation $$ \begin{align} x&=-2t+1\\ y&=2t-2 \\ z&=t \end{align}$$ What ...
1
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0answers
52 views

finding parabola equation

The centers of all circles which tangent to a circle $P(2R,0)$ with radius $R$ and to the line $x=-t$ is a Canonical parabola. Need to find the equation of that parabola and $t$ (by $R$). so: ...
4
votes
3answers
1k views

intersection of parabola and circle

Is it possible to place circle and a parabola on the plane so that their intersection consist of exactly two points, one point being a point of tangency, and the other point a transversal ...
4
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4answers
4k views

Prove analytically. The medians of a triangle intersect in a trisection point of each.

The medians of a triangle intersect in a trisection point of each.
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5answers
555 views

How can I calculate a $4\times 4$ rotation matrix to match a 4d direction vector?

I have two 4d vectors, and need to calculate a $4\times 4$ rotation matrix to point from one to the other. edit - I'm getting an idea of how to do it conceptually: find the plane in which the vectors ...
2
votes
3answers
76 views

geometry: linear transformation

I know I do it wrong but where is the mistake??? In E3* are given the points $A(1,0,0,0)$, $B(0,1,0,0)$, $D(0,0,1,0)$, $O(0,0,0,1)$ and $E(1,1,1,2)$. The linear transformation $\Phi$ operates ...
1
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0answers
94 views

Draw the locus of points which satisfy the equation

(1) Draw the locus of points $(x,y)$ which satisfy the following equation. $bx^{3}+y^{3}+x^{2}y+bxy^{2}-4abxy-2ab^{2}x^{2}-2ay^{2}+b\left( a^{2}b^{2}+a^{2}-1\right)x+\left( a^{2}b^{2}+a^{2}-1\right) ...
5
votes
3answers
2k views

What is the difference between vector components and its coordinates?

Some mathematitians told me that vector components and coordinates are different things. They say that vector $F^n$ always has N components but coordinates depend on chosen basis and, therefore, it is ...