Transformation of a cross product when multiplied by matrix A

by Math-lover1, Apr 29, 2025, 10:29 PM

I was working through AoPS Volume 2 and this statement from Chapter 11: Cross Products/Determinants confused me.

AoPS Volume 2 wrote:

A quick comparison of $|\underline{A}|$ to the cross product $(\underline{A}\vec{i}) \times (\underline{A}\vec{j})$ reveals that a negative determinant [of $\underline{A}$ ] corresponds to a matrix which reverses the direction of the cross product of two vectors.

I understand that this is true for the unit vectors $\vec{i} = (1 \ 0)$ and $\vec{j} = (0 \ 1)$ , but am confused on how to prove this statement for general vectors $\vec{v}$ and $\vec{w}$ although its supposed to be a quick comparison.

How do I prove this statement easily with any two 2D vectors?

This post has been edited 1 time. Last edited by Math-lover1, Yesterday at 10:30 PM
Reason: edit

linear algebra

matrix

vector

trigonometric functions

by VivaanKam, Apr 29, 2025, 8:29 PM

Hi could someone explain the basic trigonometric functions to me like sin, cos, tan etc.
Thank you!

trigonometry

geometry

function

Geometry books

by T.Mousavidin, Apr 29, 2025, 4:25 PM

Hello, I wanted to ask if anybody knows some good books for geometry that has these topics in:
Desargues's Theorem, Projective geometry, 3D geometry,

projective geometry

geometry

3D geometry

BABBAGE'S THEOREM EXTENSION

by Mathgloggers, Apr 29, 2025, 12:18 PM

A few days ago I came across. this interesting result is someone interested in proving this.

$\boxed{\sum_{k=1}^{p-1} \frac{1}{k} \equiv \sum_{k=p+1}^{2p-1} \frac{1}{k} \equiv \sum_{k=2p+1}^{3p-1}\frac{1}{k} \equiv.....\sum_{k=p(p-1)+1}^{p^2-1}\frac{1}{k} \equiv 0(mod p^2)}$

This post has been edited 1 time. Last edited by Mathgloggers, Yesterday at 12:22 PM
Reason: D

number theory

Sequence

by lgx57, Apr 27, 2025, 12:56 PM

$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$ . Find the general term of $\{a_n\}$ .

algebra

Sequences

Inequalities

by sqing, Apr 25, 2025, 9:19 AM

Let $a,b \in [0 ,1] .$ Prove that
$\frac{a}{ 1-ab+b }+\frac{b }{ 1-ab+a } \leq 2$ $\frac{a}{ 1+ab+b^2 }+\frac{b }{ 1+ab+a^2 }+\frac{ab }{2+ab } \leq 1$ $\frac{a}{ 1-ab+b^2 }+\frac{b }{ 1-ab+a^2 }+\frac{1 }{1+ab }\leq \frac{5}{2}$ $\frac{a}{ 1-ab+b^2 }+\frac{b }{ 1-ab+a^2 }+\frac{1 }{1+2ab }\leq \frac{7}{3}$ $\frac{a}{ 1+ab+b^2 }+\frac{b }{ 1+ab+a^2 } +\frac{ab }{1+ab }\leq \frac{7}{6 }$

This post has been edited 4 times. Last edited by sqing, Apr 25, 2025, 9:53 AM

inequalities

algebra

Geometric inequality

by ReticulatedPython, Apr 22, 2025, 5:12 PM

Let $A$ and $B$ be points on a plane such that $AB=n$ , where $n$ is a positive integer. Let $S$ be the set of all points $P$ such that $\frac{AP^2+BP^2}{(AP)(BP)}=c$ , where $c$ is a real number. The path that $S$ traces is continuous, and the value of $c$ is minimized. Prove that $c$ is rational for all positive integers $n.$

This post has been edited 2 times. Last edited by ReticulatedPython, Apr 22, 2025, 8:06 PM

geometry

inequalities

geometric inequality

Geometry Angle Chasing

by Sid-darth-vater, Apr 21, 2025, 11:50 PM

Is there a way to do this without drawing obscure auxiliary lines? (the auxiliary lines might not be obscure I might just be calling them obscure)

For example I tried rotating triangle MBC 80 degrees around point C (so the BC line segment would now lie on segment AC) but I couldn't get any results. Any help would be appreciated!

Attachments:

geometry

Three variables inequality

by Headhunter, Apr 20, 2025, 6:58 AM

$\forall a\in R$ , $~\forall b\in R$ , $~\forall c \in R$
Prove that at least one of $(a-b)^{2}$ , $(b-c)^{2}$ , $(c-a)^{2}$ is not greater than $\frac{a^{2}+b^{2}+c^{2}}{2}$ .

I assume that all are greater than it, but can't go more.

inequalities

algebra

proof-writing

N.S. condition of passing a fixed point for a function

by Kunihiko_Chikaya, Sep 6, 2009, 11:27 AM

Let $f(t)$ be a function defined in any real numbers $t$ with $f(0)\neq 0.$ Prove that on the $x-y$ plane, the line $l_t : tx+f(t) y=1$ passes through the fixed point which isn't on the $y$ axis in regardless of the value of $t$ if only if $f(t)$ is a linear function in $t$ .

function

Art of Problem Solving Blog

by Math-lover1, Apr 29, 2025, 10:29 PM

by VivaanKam, Apr 29, 2025, 8:29 PM

by T.Mousavidin, Apr 29, 2025, 4:25 PM

by Mathgloggers, Apr 29, 2025, 12:18 PM

by lgx57, Apr 27, 2025, 12:56 PM

by sqing, Apr 25, 2025, 9:19 AM

by ReticulatedPython, Apr 22, 2025, 5:12 PM

by Sid-darth-vater, Apr 21, 2025, 11:50 PM

by Headhunter, Apr 20, 2025, 6:58 AM

by Kunihiko_Chikaya, Sep 6, 2009, 11:27 AM