How to compute triple product in 2D-CodePudding

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CodePudding user response：

The vector cross triple product in 3D is

p = a×(b×c)

The full expansion can be computed with the following matrix/vector product

|px|   | -ay*by-az*bz     ay*bx        az*bx      | | cx |
|py| = |    ax*by      -ax*bx-az*bz    az*by      | | cy |
|pz|   |    ax*bx         ay*bz      -ax*bx-ay*by | | cz |

There are multiple 2D projections of the above depending if none, one or two of the vectors a, b or c are out of plane (with a z component non-zero).

In your case all three vectors are in plane (az=0, bz=0 and cz=0) which yields the following result

|px|   | -ay*by   ay*bx               0 | | cx |   | ay*(bx*cy-by*cx) |
|py| = |  ax*by  -ax*bx               0 | | cy | = | ax*(by*cx-bx*cy) |
| 0|   |      0       0    -ax*bx-ay*by | |  0 |   |                0 |

So there you have it. The right-hand side of the above is the result of a×(b×c) in 2D.