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  "body": "\n## Proof\n\nWe compare the two vectors coordinate by coordinate.\n\n1. Let $a = (a_1, a_2, a_3)$ and $b = (b_1, b_2, b_3)$. By the definition of the [[cross product|0N5N15HWZ7]], $a \\times b = (a_2 b_3 - a_3 b_2, \\; a_3 b_1 - a_1 b_3, \\; a_1 b_2 - a_2 b_1)$.\n2. The same definition with $a$ and $b$ exchanged gives $b \\times a = (b_2 a_3 - b_3 a_2, \\; b_3 a_1 - b_1 a_3, \\; b_1 a_2 - b_2 a_1)$.\n3. The first coordinate of $b \\times a$ is $b_2 a_3 - b_3 a_2 = -(a_2 b_3 - a_3 b_2)$. This is the negative of the first coordinate of $a \\times b$.\n4. The second coordinate of $b \\times a$ is $b_3 a_1 - b_1 a_3 = -(a_3 b_1 - a_1 b_3)$. This is the negative of the second coordinate of $a \\times b$.\n5. The third coordinate of $b \\times a$ is $b_1 a_2 - b_2 a_1 = -(a_1 b_2 - a_2 b_1)$. This is the negative of the third coordinate of $a \\times b$.\n6. Two vectors of $\\R^3$ are equal when their coordinates are equal. So steps 3 to 5 give $b \\times a = -(a \\times b)$.\n\n## Idea\n\nEach coordinate of $a \\times b$ has the form $x y' - y x'$, a difference of two products. Swapping the roles of $a$ and $b$ swaps the two products, and this changes the sign of the difference.\n",
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