Area of parallelogram for 4 dimensional vector

Question

Question image here

Hello, basically I just don't understand its explanation.

I also do not know what kind of formula it is used.. (It does look like it used some kinda formula)

Thank you very much if you can explain it well!!

score 2 · Answer 1 · 2016-12-13T12:55:33.410

2

Hint: Calculate the length of these two vectors and find the angle between them using inner product. After getting the height use the area formula.

edited Dec 13 '16 at 12:55

answered Dec 13 '16 at 03:13

score 2 · Answer 2 · edited Apr 13 '22 at 07:38

2

There is a theorem which states that if you have $m$ vectors in $R^n$, then the $m$-volume of the $m$-parallepiped defined by those vectors is $\sqrt{det(A^T A)}$ where $A$ is a matrix whose columns are the $m$ vectors.

In this case, $m=2$ and $n=4$.

edited Apr 13 '22 at 07:38

A. J. Pan-Collantes

2,545

answered Sep 02 '18 at 12:02

aaja2201

21

Welcome on the MathSE! Note, the site supports Latex formatting, just write $\sqrt{\det A^T}$ and you will get $\sqrt{\det A^T}$. – peterh Sep 02 '18 at 12:27
Fantastic Answer. I have verified using MS excel – Arup Hore Apr 27 '20 at 11:28
Can anybody link a reference for this formula? Thank you! – A. J. Pan-Collantes Apr 13 '22 at 06:52
Well, I have found this answer: https://math.stackexchange.com/a/1882451/601797 – A. J. Pan-Collantes Apr 13 '22 at 13:33
What is this theorem called? – Aaron Franke Aug 30 '22 at 03:31
det A^T = det A, so why bother with sqrt(x^2), why isn't the answer |det A|?? – Michael Mar 14 '24 at 21:10

score 1 · Accepted Answer · edited Feb 25 '20 at 00:13

1

This is the formula to use:

$||a||^2*||b||^2 - |a.b|^2$.

Which is the magnitude of a^2, magnitude of b^2 subtract the dot product of a.b^2.

edited Feb 25 '20 at 00:13

amWhy

210,739

answered Dec 13 '16 at 04:21

Chris

334

1

Also square root the entire equation – Chris Dec 13 '16 at 04:51
Very good answer. This formula is applicable for any 2 Vectors in any Dimension – Arup Hore Apr 27 '20 at 12:21

Area of parallelogram for 4 dimensional vector

3 Answers3