I had the same answer as you:
except I subtracted i-i = 0
Given vectors A= (Axi + Ayj), B= (Byj+Bzk) and C= (Cyj + Czk).
a) find the triple product of these vectors defined by A dot (BxC) in terms of the vector components.
I get 0 as the answer (i-i)
BxC
(Byj+Bzk) x (Cyj + Czk).
jxj is zero
jxk is i, so there is a By*Cz i term.
kxj is -i, so there is a - Bz*Cz i term
How do you on Earth get those to add to zero?
although I appreciate the comments, it seems to me there is a subtle 'note' that would have taught me that one cannot add vectors that are not on the same plane.
for example, I was subtracting your kxj is -i with the i term (which is not on the same plane or axis). That is why I got zero. If there are two negative, then I suppose it should be -2 (unless here again, I violate rule mentioned above) Thx for your help. I would also appreciate if you simply put the final answer. It is obvious, I did not understand the explanations. Thx again.
Final answer: what is it. I see -i, then I see another i. Nothing is mentioned wheter I should add like terms to get the ZERO that I god. Thx again.
To find the triple product of vectors A, B, and C defined by A dot (BxC), let's break it down step by step.
First, we find the cross product of vectors B and C:
B x C = (Byj + Bzk) x (Cyj + Czk)
To find the cross product, we can use the determinant method:
B x C = (By * Czk - Bz * Cyj)i + (Bz * Cxk - Bx * Czk)j + (Bx * Cyj - By * Cxk)k
Now, we can take the dot product of vector A with the result of the cross product of B and C:
A dot (B x C) = A dot [(By * Czk - Bz * Cyj)i + (Bz * Cxk - Bx * Czk)j + (Bx * Cyj - By * Cxk)k]
Expanding this, we get:
A dot (B x C) = (Axi + Ayj) dot [(By * Czk - Bz * Cyj)i + (Bz * Cxk - Bx * Czk)j + (Bx * Cyj - By * Cxk)k]
To find the dot product, we can distribute and combine like terms:
A dot (B x C) = (Axi * (By * Czk - Bz * Cyj)) + (Ayj * (Bz * Cxk - Bx * Czk)) + (Azk * (Bx * Cyj - By * Cxk))
Now, let's simplify further:
A dot (B x C) = (Axi * By * Czk - Axi * Bz * Cyj) + (Ayj * Bz * Cxk - Ayj * Bx * Czk) + (Azk * Bx * Cyj - Azk * By * Cxk)
Next, we can factor out the common terms:
A dot (B x C) = Axi * By * Czk - Axi * Bz * Cyj + Ayj * Bz * Cxk - Ayj * Bx * Czk + Azk * Bx * Cyj - Azk * By * Cxk
Finally, combining like terms:
A dot (B x C) = (Axi * By * Czk + Ayj * Bz * Cxk + Azk * Bx * Cyj) - (Axi * Bz * Cyj + Ayj * Bx * Czk + Azk * By * Cxk)
Since the cross product of B and C has both i and j components, and vector A only has i component, the cross product terms in the first parentheses do not have any i component. Therefore, the i component of the triple product is 0.
So, the final answer for the triple product A dot (B x C) is 0.