Algebra and Geometry | Part IA, 2004

Express the product a(a×b)\mathbf{a} \cdot(\mathbf{a} \times \mathbf{b}) in suffix notation and thence prove that the result is zero.

Silver Beard the space pirate believed people relied so much on space-age navigation techniques that he could safely write down the location of his treasure using the ancient art of vector algebra. Spikey the space jockey thought he could follow the instructions, by moving by the sequence of vectors a,b,,f\mathbf{a}, \mathbf{b}, \ldots, \mathbf{f} one stage at a time. The vectors (expressed in 1000 parsec units) were defined as follows:

  1. Start at the centre of the galaxy, which has coordinates (0,0,0)(0,0,0).

  2. Vector a has length 3\sqrt{3}, is normal to the plane x+y+z=1x+y+z=1 and is directed into the positive quadrant.

  3. Vector b\mathbf{b} is given by b=(am)a×m\mathbf{b}=(\mathbf{a} \cdot \mathbf{m}) \mathbf{a} \times \mathbf{m}, where m=(2,0,1)\mathbf{m}=(2,0,1).

  4. Vector c\mathbf{c} has length 222 \sqrt{2}, is normal to a\mathbf{a} and b\mathbf{b}, and moves you closer to the xx axis.

  5. Vector d=(1,2,2)\mathbf{d}=(1,-2,2).

  6. Vector e\mathbf{e} has length ab\mathbf{a} \cdot \mathbf{b}. Spikey was initially a little confused with this one, but then realised the orientation of the vector did not matter.

  7. Vector f\mathbf{f} has unknown length but is parallel to m\mathbf{m} and takes you to the treasure located somewhere on the plane 2xy+4z=102 x-y+4 z=10.

Determine the location of the way-points Spikey will use and thence the location of the treasure.

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