Posted by Wellness on Friday, October 30, 2009 at 10:38am.
A × B = (Ax î + Ay ĵ + Az ˆk) × (Bx î + By ĵ + Bz ˆk)
A × B = Ax î × Bx î + Ax î × By ĵ + Ax î × Bz ˆk
+ Ay ĵ × Bx î + Ay ĵ × By ĵ + Ay ĵ × Bz ˆk
+ Az ˆk × Bx î + Az ˆk × By ĵ + Az ˆk × Bz ˆk
A × B = AxBy ˆk − AxBz ĵ
− AyBx ˆk + AyBz î
+ AzBx ĵ − AzBy î
A × B = (AyBz − AzBy) î + (AzBx − AxBz) ĵ + (AxBy − AyBx) ˆk
When you dot this with C= Cxi + Cyj + Czk the dot products ii=jj=kk=1; while the dot products ij=ik=jk=kj=0.
B × C =(ByCz − BzCy) î + (BzCx − BxCz) ĵ + (BxCy − ByCx) ˆk
When you dot this with A= Axi + Ayj + Azk the dot products ii=jj=kk=1; while the dot products ij=ik=jk=kj=0.
NOTICE WHEN THIS IS DONE :
Ax(ByCz − BzCy) +Ay(BzCx − BxCz)+ Az(BxCy − ByCx) = (AyBz − AzBy) Cx + (AzBx − AxBz)Cy +(AxBy − AyBx)Cz
AxByCz - AxBzCy + AyBzCx - AyBxCz + AzBxCy - AzByCx =
AyBzCx − AzByCx + AzBxCy − AxBzCy + AxByCz− AyBxCz
So each term on the left side has an identical term on the right side of the equal sign and the identity is proven
QED
Related Questions
Vectors - Cross product of vector: If axb(cross product)=(2,0,1) and bxc= -4,0,-...
physics - For no apparent reason, a poodle is running counter-clockwise at a ...
Vectors and Geometry - if A=i+j, B=2i-3j+k and C=4j-3k find (i) (AxB)xC & Ax...
Vectors and Geometry - if A= i+j, B=2i-3j+k and C=4j-3k find (i) (AxB)xC (ii) Ax...
Help on Vectors & Geometry - (a) Given that A= 3i - j + 2k and B=2i + 3j - k...
Math - let A and B be sets. Prove that AXB=0 iff A=0 or B=0
Math Help - Ali tosses 3 number cubes,and then multiplies the results using the ...
Math - Ali tosses 3 number cubes,and then multiplies the results using the ...
math - if A and B are countable sets, then prove AxB is countable
math - if A and B are countable sets, then prove AxB is countable
For Further Reading
