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I tried looking it up but many websites just state it without proof and without intuition. I'm hoping to learn it a little bit better so that I don't forget how to compute the Jacobian when working with surface integrals where the divergence theorem is not applicable.

If you have a good online reference instead, please feel free to provide it :-)

Thanks,

User001
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    If you define the cross product by $a \cdot (b \times c) = \det(a,b,c)$ (the best definition of the cross product, IMO), and knowing that determinants represent the volumes of parallelotopes, this just comes down to Cavalieri's principle. –  Oct 06 '15 at 03:10

4 Answers4

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Choose coordinates so that the two vectors $\vec a, \vec b$ are in the $xy$-plane, with $\vec a$ along the $x$-axis. (Note that as long as you've decided on a unit length, exactly which direction you choose for the coordinate axes doesn't change anything. The vectors and their cross product live in a coordinate-free space, just floating around. We're just imposing coordinates to make concrete calculations simpler.) That means we can set $\vec a = (a_1, 0, 0)$ and $\vec b = (b_1, b_2, 0)$. This gives $$ \vec{a}\times \vec b = (0, 0, a_1b_2) $$ and the length of this vector is $\sqrt{(a_1b_2)^2} = |a_1b_2|$, obviously. But the parallelogram has base $|a_1|$ and height $|b_2|$, which means that the area of the parallelogram is given by the exact same expression.

J. W. Tanner
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Arthur
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  • Hi @Arthur - very cool and simple answer. Just a quick follow-up question - you meant for the vector b to have coordinates (0,$b_2$,0), right? Thanks, – User001 Sep 25 '15 at 22:47
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    @LebronJames No, in that case it would be established that $\vec a$ and $\vec b$ are perpendicular, which is not assumed. We know that $a_2=0$ since that's how we chose the coordinate system, but for $\vec b$ we can only tell that it lies in the $xy$-plane. – Arthur Sep 26 '15 at 08:12
  • Hi @Arthur hmm...isn't the base |a| = |$a_1$|, but then the height is |b| = $\sqrt{b_1^2 + b_2^2}$? Then the product |a||b| doesn't equal $|a_1b_1|$, which is the magnitude of the cross-product. Thanks, – User001 Sep 26 '15 at 08:30
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    @LebronJames No, the height of a parallelogram is perpendicular to the base, so $\vec b$ isn't the height. – Arthur Sep 26 '15 at 09:03
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    Ok, got it now - thanks so much for your help and patience @arthur. Have a great night. – User001 Sep 26 '15 at 10:14
  • If $\vec a$ and $\vec b$ not perpendicular? They are not base and height? – Dini Dec 27 '19 at 15:26
  • @Lifeforbetter No, they aren't necessarily perpendicular. A parallelogram isn't required to have perpendicular sides. $\vec a$ is the base, though. – Arthur Dec 27 '19 at 22:21
  • Ok. But on google image area of parallelogram is height×base and always height is perpendicular to the base...?? – Dini Dec 28 '19 at 07:22
  • @Lifeforbetter True, and $\vec a$ is the base, but $\vec b$ is not the height. – Arthur Dec 28 '19 at 09:16
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It is probably because the answer is simple in terms of classical 2D geometry.

$||\vec u\times \vec v||=||\vec u||.||\vec v||.\sin(\vec u,\vec v)$

But the area of the parallelogram defined by $\vec u$ and $\vec v$ is the base multiplied by the height. If you take $\vec u$ as the base, the height is $h=||\vec v||.\sin(\vec u,\vec v)$, hence the result...

Martigan
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    Hi @Martigan, is your equation a definition ... or can it be derived from basic geometry? If it is a derivation, do you mind just explaining a bit more about where it comes from? Thanks, – User001 Sep 25 '15 at 22:39
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$|A×B|=|A| ×|B|\sin \theta $ where $\theta$ is the angle between A and B Using simple trigonometry then we let the height of the parallelogram be a letter say X

The area of a parallelogram is given by base × height

$\sin\theta=X/|A|$ Which means that $X= |A| \sin\theta $

Therefore the area will be given by the base $|B| × |A| \sin\theta $ That is essentially the magnitude of the cross product of A and B

JMP
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Lucy
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  • @JMP Despite there's an answer which is very similar to this. I think this one made me understood better. The more you think of it is rather trivial. Unless you forget that the area of a paralellogram is the base times the height just as a rectangle. – Chris Steinbeck Bell Mar 18 '20 at 01:14
  • How is $\theta$ defined? Should this be |A x B| = |A| |B| |sin(theta)|? – NicNic8 Sep 21 '23 at 17:41
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Define $A \times_1 B, A \times_2 B$ as the vectors perpendicular to $A, B$, with orientation picked by the right hand rule, and with magnitude $|A||B|sin \theta$, respectively the absolute value of $$ \begin{vmatrix} \bf{i} & \bf{j} & \bf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} $$ A simple geometric proof shows $A \times_1 B$ is linearly dependent in $A, B$. And determinant properties show that $A \times_2 B$ is linearly dependent in $A, B$ also. It remains to observe that $\times_1, \times_2$ coincide on basis vectors $\bf{i}, \bf{j}, \bf{k}$, which means, as expected, that $\times_1 = \times_2$.

Andrei Radulescu-Banu
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