express the component of a cross product vector C= A x B in terms of Levi civita product vector and the component of a and B, using this relation show that A . ( AxB ) = 0
write a dimesions of each room as a single unit a 10' 3" * 16' 9"
Ck = eijk Ai Bj
e = 0 if any indices the same
e = 1 if order is 123, 231 etc
e = -1 if order is 132, 213 etc
so
A dot C = Ak Ck = Ak eijk Ai Bj
well for example. look at A1 C1
A1 C1 = A1 eij1 Ai Bj
= A1 A2 B3 - A1 A3 B2
now A2 C2 = A2 A3 B1 -A2 A1 B3
then A3 C3 = A3 A1 B2 - A3 A2 B1
When we add
For every PLUS we have an equal Negative
for example
A1 A2 B3 [- A1 A3 B2 ]
A2 C2 = A2 A3 B1 -A2 A1 B3
A3 C3 = [ A3 A1 B2 ] - A3 A2 B1
10' = 120 inches + 3 = 123 inches
16' = 192 inches + 9 = 201 inches
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To express the component of a cross product vector C = A x B in terms of the Levi-Civita product vector and the components of A and B, we can use the following formula:
C_i = ε_ijk * A_j * B_k
Where ε_ijk is the Levi-Civita product vector, A_j is the jth component of vector A, and B_k is the kth component of vector B. The indices i, j, and k can take values from 1 to 3, representing the three dimensions (x, y, z).
Now, let's consider the dot product A . ( A x B ). Using the formula above for the cross product, we can expand it as follows:
A . ( A x B ) = A_i * ( A x B )_i
= A_i * (ε_ijk * A_j * B_k)_i
= ε_ijk * A_i * A_j * B_k
Note that the Levi-Civita product vector ε_ijk is fully antisymmetric, meaning it changes sign if any two indices are swapped. Thus, since the components A_i and A_j are the same, we can swap them without changing the value:
A . ( A x B ) = ε_ijk * A_i * A_j * B_k
= ε_ijk * A_j * A_i * B_k
Now, let's use the antisymmetry property of the Levi-Civita product vector to rearrange the terms:
A . ( A x B ) = - ε_jik * A_i * A_j * B_k
Since the components of the Levi-Civita product vector are fully antisymmetric, ε_ijk = - ε_jik.
A . ( A x B ) = - ε_jik * A_i * A_j * B_k
But notice that this expression is exactly the same as the formula for the component of the cross product vector C = A x B that we derived earlier:
A . ( A x B ) = - C_i
Therefore, we can conclude that A . ( A x B ) = 0, as the component of the cross product vector C is perpendicular to vector A in three-dimensional space.
I hope this explanation helps! Let me know if you have any further questions.