Using properties of the triple scalar product and triple vector product,
prove that
(a ^ b) � (c ^ d) = (a � *c)(b *� d) - (a �* d)(b *� c):
[Hint: start by thinking of (a ^ b) � (c ^ d) as (a ^ b) � e, where e = c ^ d]
To prove the given equation using the properties of the triple scalar product and triple vector product, we can break down the equation step by step.
Let's start with the left-hand side of the equation: (a ^ b) � (c ^ d).
1. Begin by substituting e = c ^ d.
(a ^ b) � (c ^ d) = (a ^ b) � e
2. Now, expand the triple scalar product using the properties of the vector product:
(a ^ b) � e = (a � e) * b - (b � e) * a
3. Replace e with c ^ d in the above equation:
(a � (c ^ d)) * b - (b � (c ^ d)) * a
4. Further simplify each term using the properties of the dot product and vector product:
= ((a � c) * d) * b - ((a � d) * c) * a
5. Finally, rearrange the terms to obtain:
= (a � c) * (d * b) - (a � d) * (c * a)
This is the right-hand side of the equation, (a � *c)(b *� d) - (a �* d)(b *� c).
Thus, we have proved that (a ^ b) � (c ^ d) = (a � *c)(b *� d) - (a �* d)(b *� c) using the properties of the triple scalar product and triple vector product.