Vec & Geo
posted by Wellness .
Prove that A.(BxC) = (AxB).C

A Ã— B = (Ax Ã® + Ay Äµ + Az Ë†k) Ã— (Bx Ã® + By Äµ + Bz Ë†k)
A Ã— B = Ax Ã® Ã— Bx Ã® + Ax Ã® Ã— By Äµ + Ax Ã® Ã— Bz Ë†k
+ Ay Äµ Ã— Bx Ã® + Ay Äµ Ã— By Äµ + Ay Äµ Ã— Bz Ë†k
+ Az Ë†k Ã— Bx Ã® + Az Ë†k Ã— By Äµ + Az Ë†k Ã— Bz Ë†k
A Ã— B = AxBy Ë†k âˆ’ AxBz Äµ
âˆ’ AyBx Ë†k + AyBz Ã®
+ AzBx Äµ âˆ’ AzBy Ã®
A Ã— B = (AyBz âˆ’ AzBy) Ã® + (AzBx âˆ’ AxBz) Äµ + (AxBy âˆ’ AyBx) Ë†k
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 Ã— C =(ByCz âˆ’ BzCy) Ã® + (BzCx âˆ’ BxCz) Äµ + (BxCy âˆ’ ByCx) Ë†k
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 âˆ’ BzCy) +Ay(BzCx âˆ’ BxCz)+ Az(BxCy âˆ’ ByCx) = (AyBz âˆ’ AzBy) Cx + (AzBx âˆ’ AxBz)Cy +(AxBy âˆ’ AyBx)Cz
AxByCz  AxBzCy + AyBzCx  AyBxCz + AzBxCy  AzByCx =
AyBzCx âˆ’ AzByCx + AzBxCy âˆ’ AxBzCy + AxByCzâˆ’ 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
Respond to this Question
Similar Questions

Help on Vectors & Geometry
(a) Given that A= 3i  j + 2k and B=2i + 3j  k find A x B (b) if A= i + j, B= 2i  3j + k and C= 4j  3k find (i) (AxB)x C (ii) Ax(BxC) 
Vectors and Geometry
if A= i+j, B=2i3j+k and C=4j3k find (i) (AxB)xC (ii) Ax(BxC) thanks for your responds. 
Vectors and Geometry
if A=i+j, B=2i3j+k and C=4j3k find (i) (AxB)xC & Ax(BxC) 
Math
let A and B be sets. Prove that AXB=0 iff A=0 or B=0 
Vectors
Cross product of vector: If axb(cross product)=(2,0,1) and bxc= 4,0,4 solve magnitude of axb times magnitude of b x c a x b xbxc 
physics
For no apparent reason, a poodle is running counterclockwise at a constant speed of 3.40 {\rm m/s} in a circle with radius 2.2 {\rm m}. Let \vec v_1 be the velocity vector at time t_1, and let \vec v_2 be the velocity vector at time … 
Math
Ali tosses 3 number cubes,and then multiplies the results using the formula a x(bxc)and (axb)x c. Which property of multiplication does Ali use? 
physics
Two charges Q 1 =500/mu(C) and Q 2 =100/mu(C) are located on the XY plane at the positions r1=3/vec(j)m and r2=4/vec(i)m. Find the force exerted on the Q2(a)14.4/vec(i)+10.8/vec(j)(b)14.4/vec(i)10.8/vec(j)N(c)10.8/vec(i)14.4/vec(j)N(d)10.8/vec(i)+14.4/vec(j)N 
Physics
For the vectors with a = 3.3 and b = 2.2, what are (a) the z component of AxB, (b) the z component of AxC, and (c) the z component of BxC? 
Math
a) prove the identity: Ax(BXC)=B(A*C)C(A*B) A,B,C vectors x cross product *dot product b) use part a) to prove that cross product is not associative.