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I am wondering why nobody teached me this on universe, I always had so much trouble remembering the cross product, but now it's all so trivial, I'm talking about the following mnemonic I thought of for:

cross product

I remember the following:

  • It is defined in vector format for x, y, z (very logical step)
  • The first value (v3.x) starts with y in the calculation, then v3.y has z and v3.z follows with x.
  • Then I remember the pattern in pseudocode (if you start with y): v1.y * v2.y++ - v1.y++ * v2.y, breaking into parts:
    • The general format is 1 * 2 - 1 * 2, not hard to remember at all
    • For the vertices they go with y * y++ - y++ * y, where y++ denotes in fact z.

Does this explanation even hold? Are there facts that make sense in my world, but not in the general world? I would like to know that before I start memorizing this one, which I infact already did I think.

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4 Answers 4

up vote 1 down vote accepted

I think you got it right, but expressed it very complicated, let's look at it this way: $$ \begin{pmatrix} x\\y\\z\end{pmatrix} \times \begin{pmatrix} x'\\y'\\z'\end{pmatrix} = \begin{pmatrix} yz' - zy' \\ zx' - xz' \\ xy' - yx'\end{pmatrix}. $$ For the first row, you just remember that you have a cross $\times$ of the other two rows, starting from the top left, just as you read a book. This gives you $yz'-zy'$. The other two rows are cyclic permutations of this, so for the second row, $y$ becomes $z$, $z$ becomes $x$.

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$$ \begin{align} &(x_1,x_2,x_3)^\mathrm{T}\times(y_1,y_2,y_3)^{\mathrm{T}}\\ =&\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3 \end{vmatrix}\\ =&(x_2y_3-x_3y_2)\mathbf{i}+(x_3y_1-x_1y_3)\mathbf{j}+(x_1y_2-x_2y_1)\mathbf{k} \end{align} $$

where $(\mathbf{i},\mathbf{j},\mathbf{k})$ are orthonormal basis in $\mathbb{R}^3$.

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A much easier way of remembering: $$u=(u_1,u_2,u_3) \\v= (v_1,v_2,v_3) \\u\times v = \left|\begin{array}{ccc} i & j & k \\u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3\end{array} \right|$$

Where $i,j,k$ are the unit vectors in which the coordinates are expressed (most of the time the usual unit vectors). This comes from a property (or possible definition) of the cross product as $u\times v = w \text{ such that } \forall a\in\mathbb R^3,\ a\cdot w = \det(a,u,v)$ (using unit vector coordinates, i.e. orthonormal basis). The order which the above expression is written (what row goes where, whether it's $u\times v$ or $v\times u$, etc.) is relevant!

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$$ \begin{pmatrix} x\\ y\\ z\\ \end{pmatrix}\times \begin{pmatrix} x'\\ y'\\ z'\\ \end{pmatrix} \equiv \begin{vmatrix} \hat{\mathbf{i}}&\hat{\mathbf{j}}&\hat{\mathbf{k}}\\ x&y&z\\ x'&y'&z'\\ \end{vmatrix}= \begin{vmatrix} y&z\\ y'&z'\\ \end{vmatrix}\hat{\mathbf{i}} -\begin{vmatrix} x&z\\ x'&z'\\ \end{vmatrix}\hat{\mathbf{j}} +\begin{vmatrix} x&y\\ x'&y'\\ \end{vmatrix}\hat{\mathbf{k}}=(yz'-y'z)\hat{\mathbf{i}}+(x'z-xz')\hat{\mathbf{j}}+(xy'-x'y)\hat{\mathbf{k}}\equiv \begin{pmatrix} yz'-y'z\\ x'z-xz'\\ xy'-x'y\\ \end{pmatrix}$$

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