Questions on the use of algebraic techniques to prove geometric theorems.
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2answers
47 views
Being inside or outside of an ellipse
Let $A$ be a point $A$ not belonging to an ellipse $E$. We say that $A$ lies inside
$E$ if every line passing trough $A$ intersects $E$. We say that $A$ lies otside $E$
if some line passing trough $A$ ...
0
votes
1answer
33 views
Measuring distances on any coordinate system
I was reading the book The ABC of Relativity from Betrand Russell, and at some point, the author mentions a method for measuring the distance between 2 points on any coordinate system. He says that ...
0
votes
1answer
34 views
analytical geometry
We have an affine coordinate system and $3$ points given: $A=(1,0,0)$, $B=(0,1,0)$, $C=(0,0,1)$, $D=(1,1,1)$. I have to find a linear transformation, which depicts the points $A$, $B$, $C$ and $D$ ...
0
votes
1answer
94 views
Geometric question?
First of all, is it Geometric?
Image of the drafted:
I need help solving this question, and I am completely lost on how can I solve this.
Could anyone explain the way of solving this geometric ...
0
votes
0answers
25 views
How to introduce perpendicular or congruence of angles in affine space
$n$-dimensional affine point-vector space is a pair $\mathbb A^n = \langle \mathbb A, V^n \rangle$, where $\mathbb A$ is an arbitrary set, which elements are called points of affine space, $V^n$ is an ...
0
votes
1answer
33 views
triangle, vectors, proving an identity.
I'm trying to prove something but unfortunately I can't.
Let $ABC$ be a triangle and $M$ a point in $[AB]$ where $d(A,M)=d(B,M)$.Let also be
$N$ be a point in $[AC]$ where $d(A,N)=d(B,N)$.
Prove ...
0
votes
1answer
30 views
Expressing a point in two coordinate systems
Let $(O,e_1,e_2,e_3)$ and $(O',e_1',e_2',e_3')$ be two coordinate systems. Let $\overline{OO'}=2e_1-e_2+3e_3$, $e'_1=e_1-e_2+3e_3$, $e'_2=e_1+e_2+e_3$ and $e'_3=e_1-e_2-e_3$.
a) Find the coordinates ...
0
votes
1answer
29 views
Get the equation for a plane when we know a point and an intersection between two planes
The point is $P:(1,4,-2)$ and the two planes that the equation intersect is $$\pi_1:2x+2y-z+4=0$$$$\pi_2:3x-y+3z+1=0$$ what is the equation?
1
vote
1answer
29 views
Find equations of the ellipses given conditions on the directrices, foci, and vertices
The ellipses have their centers at the origin and their major axes on the $x$-axis. Find the equation:
with distance between directrices $27$, and between foci $3$;
with a focus at $(-\sqrt{13},0)$ ...
-1
votes
0answers
26 views
A problem of Analytical Geometry.
Lines are drawn through the origin to meet the circle in which plane $x+y+z=1$ cuts the sphere $x^2+y^2+z^2-4x-6y-8z+4=0$. How to show that they meet the sphere again at points on the plane $y+2z=2$?
1
vote
1answer
20 views
Line equation - parametric and canonical
Let's say I have a line in R3:
$$
l:\begin{cases}
x-3y+3z=0\\
x+2y-2z=2
\end{cases}
$$
How to change it to canonical and parametric equation?
2
votes
2answers
37 views
To use the two-point formula to find the linear equation relating $C$ and $F$:
I've tried to solve a problem which I'm going to give below. What I don't understand is that which variable is dependent and which is independent among $C$ and $F$. I think we can relate $C$ and $F$ ...
2
votes
3answers
51 views
To find the x and y-intercepts of the line $ax+by+c=0$
Please check if I've solved the problem in the correct way:
The problem is as follows:
Find the points at which the line $ax+by+c=0$ crosses the x and y-axes. (Assume that $a \neq 0$ and $b \neq ...
3
votes
4answers
54 views
Equation of the line that has $x$ and $y$ intercepts at $a$ and $b$.
Please can anyone help me with proving the following problem:
Show that the line that crosses the $X$-axis at $a \neq 0$ and the $Y$-axis at $b \neq 0$ has the equation $$\dfrac{x}a + \dfrac{y}b ...
2
votes
1answer
46 views
Proof that differential of differential form $=0$ i.e $d(df) = 0$.
Let $f$ be a differentiable function on an open space $U \subset \mathbb{R}^n$. Proove that $d(df) = 0$.
So my proof is:
Let
$$f = \sum c_{i_1, \cdots i_k}(x_*)dx_{i_1} \wedge \cdots ...
0
votes
1answer
23 views
Prove that the $2$ form defines a symplectic structure
Prove that the $2$ form
$$\omega = -2[(1+x_2^2)dx_1 \wedge dx_2 + dx_1 \wedge dx_3 + dx_3 \wedge dx_4]$$
defines a symplectic structure on $\mathbb{R}_x^4$.
My definition of as ...
4
votes
0answers
106 views
Are there eigenvectors, eigenvalues, and characteristic functions for non-linear vector fields?
An eigenvector is a vector in the preimage of the transformation whose direction is not changed when the transformation is applied. It seems like the concept of eigenvectors and eigenvalues would ...
1
vote
2answers
47 views
How can we derive the standard form of the linear equation: $Ax+By+C$?
How can we derive the standard form of the linear equation: $Ax+By+C=0$? What do "$A$", "$B$" and "$C$" in the standard form of the linear equation mean? As in the point-slope form of the linear ...
0
votes
2answers
36 views
Find the equation of a locus…(Read More)
Find the equation of the locus of a point which moves so that it's distance from (4,-3) is always one-half its distance from (-1,-1).
0
votes
3answers
30 views
Identify the curve with the following equation.
To "identify" means not only to name but to give pertinent data, such as center, foci and axes, if they exist.
$$4x^2=4x-y^2$$
0
votes
2answers
32 views
Local Diffeomorphism Theoerm
Is this correct for the local diffeomorphism theorem:
A multivariable function $F(x_1, \cdots x_n)$ has a local diffeomorphism at a point $a = (a_1, \cdots a_n)$ if the determinant of the Jacobian ...
0
votes
0answers
51 views
Derivation of $Ax+By+C=0$
Below I've tried to derive the standard form of the linear equation. Please check my derivation.
Object: Derive $Ax+By+C=0$.
The slope-intercept form of the linear equation can be given as:
...
-1
votes
1answer
30 views
Solve a parabola problem?
So, two parabolas are given: $y^2=24x$ and $x^2=3y$ and a point $A(24,3)$. If B and O are intersect points of these two parabolas, prove that the angle ABO is right.
1
vote
1answer
36 views
Coordinates and locus of centroid
A triangle has two of its sides along the co-ordinate axes and its third side is a tangent to the circle $x^2+y^2=a^2$. If the coordinates of the point of contact of the tangent are $(a \cosØ,a ...
0
votes
1answer
42 views
How can you use the parametric form of a plane in the following application?
With regards to computer graphics, how can you make use of the parametric representation of a plane and how of its normal form?
1
vote
1answer
43 views
Willmore energy of an ellipsoid
Given an ellipsoid of equation
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$
How can I calculate the Willmore energy of this surface knowing that its definition is:
...
1
vote
1answer
59 views
Triangle optimization problem
Let $a,b,c$ be the sides of a triangle , then what is the maximum and minimum values (if exist) of the following quantities
(i) $\dfrac {a^2b^2c^2}{(a+2b)(a+2c)(b+2c)(b+2a)(c+2a)(c+2b)}$
(ii) ...
0
votes
0answers
41 views
How to prove the property of scalar distribution over vector addition when the vectors are collinear?
$\overrightarrow{a},\overrightarrow{b} \in V^3 , \alpha \in \mathbb{R} $
Prove: $\alpha(\overrightarrow{a} +\overrightarrow{b}) = \alpha\overrightarrow{a} + \alpha\overrightarrow{b}$
When $\alpha = ...
0
votes
0answers
71 views
Overlapping Areas
Knowing the areas of A, B and C, is there an analytical way to find out if two arbitrary shapes overlap each other in the plane? (See image here: http://i.imgur.com/RWsqysT.jpg)
More formally, I'm ...
1
vote
4answers
42 views
find vector reflected across another vector
I apologize that this may sound like a very basic question, but I can't find any clear answers in my search. I have a vector $\vec{v}$ that I want to reflect across a vector $\vec{n}$. The dot product ...
3
votes
2answers
53 views
Slope; A measure of Direction
In my book, the definition of the slope of the straight line is:
The slope is a measure of the direction of the line.
1) When the line has no slope, it tells that it is vertical or moving ...
1
vote
1answer
40 views
Understanding the graph of the displacement of a particle wirh respect to time
At time $t=0$ the position of the particle is $3 ft$, and at time $t=2$ the position of the particle is $11ft$. At time $t=0$ the velocity of the particle must have been zero. So if its the motion ...
2
votes
3answers
65 views
Analytic geometry straight line problem
Prove that two straight lines represented by the equation $x^3+y^3+bx^2y+cxy^2=0$ will be at right angles if $b+c=-2$
I didn't know that even straight lines like planes can be represented by a ...
0
votes
2answers
28 views
Analytic geometry section of cone and sphere
How to show that the cone $yz+zx+xy=0$ cuts the sphere $x^2+y^2+z^2=a^2$ in two equal circle ?
I understand that the two equations taken together represent the circle. but how to go about finding the ...
1
vote
3answers
78 views
surface area of a sphere above a cylinder
I need to find the surface area of the sphere $x^2+y^2+z^2=4$ above the cone $z = \sqrt{x^2+y^2}$, but I'm not sure how. I know that the surface area of a surface can be calculated with the equation ...
3
votes
1answer
25 views
Point-slope Equation
Let suppose that $R(x_1, y_1)$ is a point on the (x, y)plane and a line $L$ with slope $m$ passes through this point. There is a point $S(x_1, y_1)$ on $L$ such that $R$ and $S$ are coincident points. ...
0
votes
2answers
61 views
Straight Line Definition
Definition: Let $P(x_0, y_0)$ be a point and let $m$ be a real number. The line through $P$ with slope $m$ is the set of all points $Q(x, y)$ with,
$y -y_0 = m(x - x_0)$
Does the set of all ...
0
votes
2answers
32 views
Slope of a straight line
Why is this so that a higher value of slope indicates a steeper incline? I can't take it into my head. What could be the reason behind that? I know that it is a fact because I've also noticed it but ...
0
votes
1answer
27 views
Diameter and Hausdorff Distance
Let $A,B \subset \mathbb R^n$ be non empty compact sets and $d_H$ be Hausdorff distance. I'm thinking that if we know the distance between two sets, the difference between their diameters is bounded. ...
0
votes
0answers
29 views
analytical geometry - volume of tetrahedron
Can someone please explain to me the meaning of this statement in relation to calculating the volume of tetrahedron ?
"We shall often get negative value for volume of a tetrahedron. We define the ...
1
vote
2answers
24 views
given coordinates of beginning and end of two intersecting line segments how do I find coordinates of their intersection?
There are two line segments. I know for sure they intersect (so I don't have to check it). For both line segment I know coordinates of its both ends. With what formula can I find coordinates of their ...
11
votes
2answers
294 views
When do equations represent the same curve?
Suppose we have two sets of parametric equations $\mathbf c_1(u) = (x_1(u), y_1(u))$ and $\mathbf c_2(v) = (x_2(v), y_2(v))$ representing two 2D planar curves. When I say "2D planar curves" I mean ...
1
vote
1answer
33 views
What is the dimension of the tangent space
Write down the equations of the tangent space $T_aX$ to a submanifold $X = \{f_1 = \cdots = f_k = 0 \}$ backwards $\in a = \{a_1, \cdots a_n \}$. What is the dimension of $T_aX$.
I wrote the ...
0
votes
1answer
24 views
determining if a curve passes through a loop
I have two curves described by parametric equations, and one is a closed loop. How do I analytically determine whether or not the other curve passes through the loop? That is, without graphing and ...
1
vote
2answers
89 views
Finding the locus of the midpoint of chord that subtends a right angle at $(\alpha,\beta)$
There is a circle $x^2+y^2=a^2$. On any line that cuts the circle in two distinct points(it is a secant), the points of intersection with circle are taken and at those two points I draw the tangents ...
0
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0answers
77 views
What is this expression called?
Could anyone please tell me if they recognize this equation? What it does is calculate the angle between two lines, but I need it's name. Any help is greatly appreciated!
$$\sin \theta = A_{1} \cdot ...
0
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1answer
44 views
Definition of a submanifold
Say what it means for a system of equations
$$f_1 = \cdots = f_m = 0,$$
where $f_i(x_1, \cdots x_n)$ are differentiable functions, to define a submanifold near a point $a = (a_1, \cdots ...
0
votes
0answers
24 views
I get the value of my $2$ form to be $0$.
Let $\omega = (x_1 + \cdots +x_n) \sum_{i<j} dx_i \wedge dx_j$ be a $2$-form on $\mathbb{R}^n, \omega \in \Omega (\mathbb{R}^n)$. Compute the function
$$\omega \left( ...
0
votes
1answer
30 views
Sketching the circle for the equation: $\sqrt{(x -1)^2 + (y-1)^2} = \sqrt{2}$
How should I sketch the circle for the equation mentioned in the title? If I calculate the square root of the number $2$ it continues to infinity.
$\sqrt2 = 1.414213562...$
1
vote
1answer
33 views
Determining the Diameter of a Circle
The radius of a circle can in one way be determined if we know its Diameter $D$. It can be determined by the equation:
$r = \frac D2$ ..... (I)
We can also determine the radius of a circle by ...
