Tagged Questions

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

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Alternate form for the vector field $x^3\cdot\hat{x} + y^3\cdot\hat{y}$

Is there an alternate form for the vector field $x^3\cdot\hat{x} + y^3\cdot\hat{y}$ in which we can write all in function only of the radius $r=\sqrt{x^2 + y^2}$ ? Thank you
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How to prove the parallel projection of an ellipsoid is an ellipse?

Take the following ellipsoid in implicit form as an example: $$x^2 + 2 y^2 + 3 z^2 + x y + y z - 2 xz = 5$$ which shows: The parallel projection of the ellipsoid onto $xoy$ coordinate plane can ...
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Great mathematical fusions in math history

Development of the mathematics resembles usually a growing tree - from old branches grow new ones. However sometimes domains of mathematics which were separated for the long time are fused together ...
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Find the centre of a circle circumscribing the triangle whose angular points are $(1,1), (2,3), (-2,2)$

The main question is as follows : Find the point $P$, such that $P$ is the centre of a circle circumscribing the triangle whose angular points are $(1,1), (2,3), (-2,2)$. My method : I am ...
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Mathematical properties of two dimensional projection of three dimensional rotated object

Please be gentle as I do not have any degree in maths. By using a compass/straighedge method to construct Metatron's cube, a regular dodecahedron can be inferred from intersecting points. I'm looking ...
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Find the coordinates of P and Q.

A line is drawn through the point $A(1,2)$ to cut the line $2y=3x-5$ in $P$ and the line $x+y=12$ in $Q$. If $AQ=2AP$, find the coordinates of $P$ and $Q$. From: Mathematics, The Core Course for A-...
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Is there a simple way to decide if a hyperboloid is one-sheeted or two-sheeted, given the quadric equation?

Let us say that we have a quadric equation, whose solution set lies in $\mathbb{R}^3$, and you know it's a hyperboloid. Is there a way to analytically decide through a criterion if the hyperboloid is ...
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Checking nature of angles of a triangle given the equations of the three lines that form a triangle

Suppose we have three lines $\ell_i=a_ix+b_iy=c_i$, $i=1,2,3$ and we are given that they form a triangle. I need to find which angles are acute and which are obtuse without plotting the lines ...
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Clarification Needed:Equation Of Refracted Ray/Line

A ray of light is sent along the line $2x-3y=5$.After refracting across the line $x+y=1$ it enters the opposite side after turning by $15^0$ away from the line $x+y=1$.Find the equation of the ...
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Given $\angle (\vec{u} , \vec{v}) = 30^\circ$, $\Vert \vec{w} \Vert = 4$ and $\vec{w}$ is $\perp$ to both find $\vec{u} \cdot \vec{v} \times \vec{w}$

I am given the following problem: Knowing that the angle between the unit vectors $\angle (\vec{u} , \vec{v}) = 30^\circ$, $\Vert \vec{w} \Vert = 4$ and that $\vec{w}$ is orthogonal to both of ...
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Using cross product prove that if $\vec{u} \times \vec{v} = \vec{0}$ and $\vec{u} \cdot \vec{v} = 0$ then $\vec{v} = \vec{0}$

I am asked to elaborate on the following proof: Let $\vec{u} \neq \vec{0}$. Prove that if $\vec{u} \times \vec{v} = \vec{0}$ and $\vec{u} \cdot \vec{v} = 0$ then $\vec{v} = \vec{0}$. My attempt ...
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False proof: $\pi = 4$, but why?

Note: Over the course of this summer, I have taken both Geometry and Precalculus, and I am very excited to be taking Calculus 1 next year (Sophomore for me). In this question, I will use things that I ...
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Prove that locus of orthocentre is $x+y=0$

The base of a triangle is axis of $x$ and its other $2$ sides are given by the equations :$y = (1+α)\frac{x}{α} + (1+α)$ and $y = (1+β) \frac{x}{β} + (1+β)$.Prove that the locus of its orthocentre is ...
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Pair of straight lines problem: Prove that $g (a_1+b_1)=g_1 (a+b)$

If the lines joining the origin and the point of intersection of the curves $ax^2+2hxy+by^2+2gx=0$ and $a_1x^2+2h_1xy+b_1y^2+2g_1x=0$ are mutually perpendicular then prove that $g (a_1+b_1)=g_1 (a+b)$...
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Get the equation of a circle when given 30 points [closed]

A similar question has been asked before on this site but that was of getting equation of circle using 3 points. I want my center to be more accurate So my question is how can i get the center of ...
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2D AABB count vector for not having collision after movement

I hope it is correct here, I feel like this question is more math related than programming. Table of Contents Introduction What is question about / problem description What way I figured out Other ...
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Condition for n points in the plane to determine a convex n gon

Suppose there are n points in the plane, labelled 1 through n, no three of which lie on a line. Suppose further that for every triple [i,j,k] with i< j < k that travelling from i to j to k is ...
I am working on an forward kinematics geometric solution. What it roughly means is that given a certain $(x,y,z)$ co ordinates the robot should compute the angles at which it should move its arms. I ...
I am given the following exercise: Show that $\Vert \overrightarrow{a} + \overrightarrow{b} \Vert = \Vert \overrightarrow{a} \Vert + \Vert \overrightarrow{b} \Vert$ if and only if \$\...