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henryx

Vector maths

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henryx    128
I''m not sure about the expansion of a vector dot product and am wondering if any of you maths people can help. I have four vectors v1, v2, v3, v4 and want to get the dot product (v2-v1).(v4-v3) Can anybody tell me if this dot product can be expanded to: v2.v4 - v2.v3 - v1.v4 + v1.v3 I''ve tried this out with numbers and my feeling is that it isn''t a valid expansion - any help would be appreciated on whether this is valid or not, and if not what the correct expansion is (if there is one)? thanks henry

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Enigma    1410
OK, lets see:
take - v1 = (a1i , a2j , a3k ), v2 = (b1i , b2j , b3k ), v3 = (c1i , c2j , c3k ), v4 = (d1i , d2j , d3k ).

v2 - v1 = ((b1 - a1)i , (b2 - a2)j , (b3 - a3)k )

v4 - v3 = ((d1 - c1)i , (d2 - c2)j , (d3 - c3)k )

(v2 - v1 ).(v4 - v3 ) = (b1 - a1)(d1 - c1) + (b2 - a2)(d2 - c2) + (b2 - a2)(d2 - c2)
(v2 - v1 ).(v4 - v3 ) = b1d1 - b1c1 - a1d1 + a1c1 + b2d2 - b2c2 - a2d2 + a2c2 + b3d3 - b3c3 - a3d3 + a3c3

v2 .v4 - v2 .v3 - v1 .v4 + v1 .v3 = (b1d1 + b2d2 + b3d3) - (b1c1 + b2c2 + b3c3) - (a1d1 + a2d2 + a3d3) + (a1c1 + a2c2 + a3c3)
v2 .v4 - v2 .v3 - v1 .v4 + v1 .v3 = b1d1 - b1c1 - a1d1 + a1c1 + b2d2 - b2c2 - a2d2 + a2c2 + b3d3 - b3c3 - a3d3 + a3c3
v2 .v4 - v2 .v3 - v1 .v4 + v1 .v3 = (v2 - v1 ).(v4 - v3 )

So they are identical.

EDIT: a few bad tags

Enigma
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...weight, or fail to invite them out...

Edited by - Enigma on December 13, 2001 6:34:56 AM

Edited by - Enigma on December 13, 2001 6:35:49 AM

Edited by - Enigma on December 13, 2001 6:39:49 AM

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