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

learn more… | top users | synonyms (1)

1
vote
1answer
60 views

How can I find the common axis of 2 cones in space that have the same base radius but different heights?

How do I find the 3D vector describing the axis of 2 overlapping cones, like this: If I have only the following information: Coordinates of the common tip Coordinates of a point on the yellow ...
1
vote
2answers
23 views

Can ellipse equation be transformed through one of its foci?

Can we transform ellipse equation to represent an ellipse transformed by tilting it through its focus such that its center point moves in circular manner and one of its focus stays at constant ...
1
vote
2answers
41 views

Is it possible to find equation for ellipse when focus, eccentricity and two points are known?

Is it possible to find equation for an ellipse when we know two points and one focus in 2d cartesian coordinate system? We can also make these assumptions about these two given points depending on ...
10
votes
3answers
9k views

Equation of angle bisector, given the equations of two lines in 2D

I have two lines in 2D expressed with general equation (or implicit equation): First line: $a_1x+b_1y=c_1 \qquad(1)$ Second line: $a_2x+b_2y=c_2 \qquad(2)$ If the two lines are intersecting I will ...
2
votes
0answers
14 views

Find a vector equation and parametric equations for the line segment that joins $P$ to $Q$. [duplicate]

Find a vector equation and parametric equations for the line segment that joins $P$ to $Q$. Here $P(1,-1,7)$ and $Q(7,5,1)$. I have tried to find $r(t)$ by using the formula $r(t)=p+t(p-q)$ but ...
0
votes
2answers
36 views

What does 'forms a right-handed set' mean?

In a question I am reading, the following question appears. What if $\vec{A},\vec{B}$, and $\vec{C}$ are mutually perpendicular and form a right-handed set? What exactly does "form a ...
-1
votes
3answers
37 views

point on a line and distance from a point [closed]

I have point(x1,y1) and point(x2,y2) these are end point of line and point(m,n) is a point. How can i find Point(a,b) which lies on the line ,that is the shortest path from point(m,n) to the line
1
vote
1answer
391 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 ...
0
votes
2answers
16 views

Why $(h,k)$ in equation $y= a(x-h)^2 +k$ is the vertex of a parabola?

As in the title , I know how to convert normal explicit equation to a vertex form equation by completing the square . But what is the reasoning behind why $(h,k)$ must be the vertex , but not other ...
2
votes
4answers
74 views

Find the equation of a circle which is tangent to $y$-axis at a given point and cuts a chord of given length on the $x$-axis

How to find the equation of the circle which touches $y$ axis at $(0,3)$ and cuts a chord of length $8$ on the $x$ axis? It should look like this: My approach: Since the circle touches $y$ ...
1
vote
2answers
26 views

Hyperbola question

the graph $ y^2=16x $ is a hyperbola; it can be rewritten as $ y= \pm 4\sqrt{x}$ when I draw it down however It is clearly not a function..question is whether it has to be one in order to perform ...
0
votes
2answers
44 views

Finding the equations of the lines and tangent to the circle

Find the equations of the lines through $(2,0)$ and tangent to the circle $x^2+y^2=1$. I tried to solve this and I know the right answer but I just can't solve this. The right answer: $\sqrt{3}y=x-2$ ...
3
votes
1answer
317 views

$2D$ Line Segment - Triangle Intersection

I've seen similar questions but could not solve my problem with those. My question is how to detect an intersection of a line segment and a triangle on a 2D coordinate system? I don't need the point ...
2
votes
1answer
187 views

Intersection of 3D curves parameterised by piecewise defined functions

I need to calculate the intersection of two 3D parametric curves $\vec{C_1}$ and $\vec{C_2}$. Those curves are parameterised by piecewise functions. $\vec{C_1}= ...
3
votes
1answer
48 views

How to compute coordinates of a point that intersects an sphere

Hi all. Is there a way to compute the S(x,y,z), given the following information: A(x,y,z) e = elevation (from the line AS) Az = azimuth (over A). Perpendicular to x axis. Can vary from 0 tp 360. ...
0
votes
0answers
36 views

Unit sphere x axis intersections

This is a problem from a vector calculus textbook, Higher Order Derivatives Consider the unit sphere S given by $x^2+y^2+z^2=1$. S intersects the x-axis at 2 points. Which variables can we solve for ...
0
votes
1answer
14 views

Hyperbola / Rotated Hyperbola Intersection

I am trying to find the point where two hyperbolas intersect, that is, to find a vertex that is common to both hyperbolas. Also, note that I am only testing for a region of both hyperbolas -- only a ...
1
vote
2answers
341 views

I want to find 3 planes that each contain one and only one line from a set

The three lines intersect in the point $(1, 1, 1)$: $(1 - t, 1 + 2t, 1 + t)$, $(u, 2u - 1, 3u - 2)$, and $(v - 1, 2v - 3, 3 - v)$. How can I find three planes which also intersect in the point $(1, 1, ...
-1
votes
1answer
649 views

Ray VS Line intersection [closed]

Continue vector vs plane intersection How to determine Ray (one point R(rx;ry) and alpha with OX) with line (two points A(ax;ay) and B(bx;by)) intersection?
2
votes
0answers
15 views

About parametric equation of a line in $3$-space

$a.$ Given coordinates $(x, y, z )$ with origin $(0,0,0)$, parameterize the line through the points $(4,5,6)$ and $(1,2,3).$ $b.$ Take components of your answer to Part $(a)$ to give three ...
0
votes
1answer
15 views

What is the vector equation of the line through the head of $v_0$ and parallel to $v_p$?

$v_0$ and $v_p$ are vectors. Let $v_0, v_1$ and $v$ be vectors, all emanating from $(0, 0, 0)$. Suppose the line $l$ is passing through their heads. Let $v_p$ be on the line $l$ such that $v_1 = v_0 ...
8
votes
2answers
9k views

Finding the intersecting points on two circles

Given 2 circles on a plane, how do you calculate the intersecting points? In this example I can do the calculation using the equilateral triangles that are described by the intersection and centres ...
0
votes
1answer
22 views

Simple Analytic Geometry Problem [closed]

Could someone explain to me the solution of this problem? Find the equation of a circle passing through (3,7) and tangent to the lines x-3y +8=0 and y=3x
1
vote
2answers
136 views

Using the Invariant Principle to prove a coordinate can't be reached

Problem: A robot wanders around a 2-dimensional grid. Starting at $(0, 0)$, he is allowed 4 different kinds of steps: $(+2, -1)$ $(+1, -2)$ $(+1, +1)$ $(-3, 0)$ He is trying to get to $(0, 2)$. ...
0
votes
0answers
28 views

Endless repeating tiles

Regarding to this question ( "Hall of mirrors" OR "wraped plane" - Problem ) there's still an open point: how can i determine if the sum of all vectors pointing from p1 to p2 is ...
-1
votes
2answers
15 views

finding the ratio which divides the segment

Find the ratio in which the point (2,-1) divides the segment from (6,1) to (0,-2). Find the coordinates of the point that divides the segment from (0,-1) to (6,3) I the ratio 2:5 . can somebody ...
1
vote
2answers
21 views

How to find the equation of $f(x^3)$ given the tangent in $f(x)$? [closed]

The exercise says: The tangent of $f(x)$ in $x=1$ is $y=2x-1$. Find the tangent of $y=f(x^3)$ in $x=1$.
0
votes
2answers
47 views

About the sum of sines of two angles

Suppose that $0\le \alpha\le \pi/2$ and $0\le \beta\le \pi/2$ such that $\alpha+\beta\ge \pi/2$. Can we prove that $\sin(\alpha)+\sin(\beta)\ge 1$?
2
votes
1answer
19 views

finding a soild from five planes

Given five planes: $\pi_1=2x+5y+z-2=0$ $\pi_2=x+y-z-1=0$ $\pi_3=x+4y+2z-4=0$ $\pi_4=3x-y+4z-3=0$ $\pi_5=-6x+2y-8z+k=0$. How can i find the solid shape that is formed by those planes? I tried to ...
2
votes
1answer
43 views

Why ternary diagrams work

I am trying to understand why ternary diagrams work. In order that the altitude criterion be valid, if I correctly understand, given equilateral triangle $ABC$, whose vertices I name as the three ...
6
votes
4answers
230 views

Is there a name for the curve $t \mapsto (t,t^2,t^3)$?

Is there a name for the curve given by the parametrization $\{(t,t^2,t^3); t\in\mathbb R\}$? Here is a plot from WA. An another plot for $t$ from $0$ to $1$. This curve is an example of a ...
1
vote
3answers
40 views

Describe geometry of complex plane.

Let $a \in \Bbb R$ and $c>0$ to be fixed. Describe the set of points $z$ such that $|z-a|-|z+a|=2c$ for every possible choice of $a$ and $c$. Then let $a$ be a complex number using the rotation ...
1
vote
2answers
57 views

Proving $proj_{proj_{\vec u} \vec v} \vec v=proj_{\vec u} \vec v$

Can anyone show me how to prove: $proj_{proj_{\vec u} \vec v} \vec v=proj_{\vec u} \vec v$? I got confused trying to prove it (not geometrically)... Thanks in advance!
2
votes
2answers
26 views

Equation of tangent from a point outside it

Is there any general method of finding the equation of the tangent of a function $f(x)$ from a point $(a,b)$? $\hspace{1 mm}$ Then how do you find the angle between two tangents from $(0,0)$ to a ...
2
votes
3answers
77 views

Coordinate of the excentre of a triangle

I am just wondering that how the coordinate of the excentre comes out if we know the coordinates of vertices of the triangle.
1
vote
1answer
33 views

Why does $(a-2b)\times (3a+2b) = a\times (3a+2b) - 2b \times(3a+2b)$?

Let $\textbf{a},\textbf{b}\in\mathbb{R}^3$ be such that $|\textbf{a}| = |\textbf{b}|$ and the angle between them is $45º$. We had a test where we should find the answer of ...
1
vote
1answer
20 views

Point coordinates at a fixed distance from a vector

I would like to solve the following generic problem by using vector notation that I will use it to improve my algorithm. I have a vector P1P2 that points P1 and P2 are known. Furthermore, an ...
0
votes
2answers
54 views

Change of basis matrix for polynomials?

I've understood what a change of basis matrix is, and how it's structured. So a change of basis matrix from $B$ to $C$ is the matrix $M$ such that: $${\begin{bmatrix} &\\ \\ \\\end{bmatrix}}_B ...
0
votes
1answer
371 views

Find locus of points relating to an ellipse

I would like to find the equation of the following locus. For a big circle C centered at (0,0), the locus of points that the sum of distances to Y-axis and to C is 1, say in the first quadrant, is ...
2
votes
6answers
86 views

Intersection of a sphere and a plane

How can I find the intersection between the sphere $x^2+y^2+z^2=1$ and the plane $x+y+z=1?$ Context This is related to a computation of surface integral using Stokes' theorem, Calculate the surface ...
3
votes
1answer
25 views

Focal length of an ellipse and related results

There are 2 questions(part of same question but I divided it into two): Q1. Prove that the length of the focal chord of the ellipse $\frac {x^2}{a^2}+\frac {y^2}{b^2}=1$ which is inclined to the ...
0
votes
2answers
43 views

Finding midpoint of rectangle in 3D vectors

If given the points (-10,-2,0), (-10,2,0), (-12,0,2) and (-12,0,-2), how do I find the midpoint?
1
vote
3answers
292 views

Arc Length Formulas

Use the arc length formula to find the arc length of the upper half of the circle with center at $(0,0)$ and radius $3$. Also, find the arc length of the curve in the first question by using ...
0
votes
1answer
28 views

Find the map of the closed ball $B(0,1)$ of the following continuous function $f(x,y,z)=(\frac x3,\frac y2-1,\frac z9+1)$ and $f^{-1}(0)$.

Find the map of the closed ball $B(0,1)$ of the following continuous function $$f(x,y,z)=\left(\frac x3,\frac y2-1,\frac z9+1\right)$$ and $f^{-1}(0)$. $f^{-1}$ seems quite simple, I got ...
0
votes
1answer
26 views

Plotting 3 equidistant points on a sphere

.Hello! I'm trying to figure out how to plot with x,y,z, three points that are equidistant along the surface of a sphere from each other that are all on a horizontal axis (so y = 0) with a radius of ...
0
votes
2answers
380 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?
3
votes
3answers
41 views

The locus of points $z$ which satisfy $|z - k^2c| = k|z - c|$, for $k \neq 1$, is a circle

Use algebra to prove that the locus of points z which satisfy $|z - k^2c| = k|z - c|$, for $k \neq 1$ and $c = a + bi$ any fixed complex number, is a circle centre $O$. Give the radius of the circle ...
3
votes
1answer
46 views

Motivation for constructing $F$ s.t. $\ker(\text{curl}) \subset \text{Im}(\text{grad})$, $\ker(\text{div}) \subset \text{Im}(\text{curl})$

In 'from calculus to cohomology', we consider the space $V$ of smooth functions $U \to R^3$, with $U \subset R^3$ star-shaped (i.e. convex), and for cohomology reasons (showing $H^1(U)=H^2(U)=0$) we ...
1
vote
1answer
28 views

Area of surface parametrized in spherical coordinates

Suppose we have a smooth, bounded, closed surface in $\mathbb{R}^3$ which can be parametrized by giving the distance from the origin as a function $r(\varphi,\theta)$ of spherical angles ...
3
votes
3answers
99 views

3D coordinates of circle center given three point on the circle.

Given the three coordinates $(x_1, y_1, z_1)$, $(x_2, y_2, z_2)$, $(x_3, y_3, z_3)$ defining a circle in 3D space, how to find the coordinates of the center of the circle $(x_0, y_0, z_0)$?