Show that if a, b, c are in V3, then
(a x b) dot [(bxc)x(cxa]) = [a dot (b x c)]^2
I don't know what properties to use
To prove the given equation involving vectors, we can start by expanding both sides and simplifying the expressions.
Let's begin by expanding the left-hand side (LHS):
(a x b) dot [(b x c) x (c x a)]
Using the properties of the dot product and triple cross product, we can rewrite this expression as:
(a x b) dot [(b x c) x (c x a)]
= (a x b) dot [(b dot (c x a)) - (c x a) dot b]
= [(a x b) dot (b dot (c x a))] - [(a x b) dot (c x a dot b)]
Next, let's expand the right-hand side (RHS):
[a dot (b x c)]^2
Using the properties of the dot product and cross product, we can rewrite this expression as:
[a dot (b x c)]^2
= [(a dot (c x b))]^2
= [(a dot (-b x c))]^2
= [-(a dot (b x c))]^2
= [- (a dot (b x c))^2]
Now, let's compare the LHS and RHS expressions:
LHS = [(a x b) dot (b dot (c x a))] - [(a x b) dot (c x a dot b)]
RHS = - (a dot (b x c))^2
Since the LHS is equivalent to the negative of the RHS, we can rewrite the equation as:
[(a x b) dot (b dot (c x a))] - [(a x b) dot (c x a dot b)] = - (a dot (b x c))^2
Now, to simplify the LHS expression further, we can consider the properties of the dot product and cross product:
(a x b) dot (b dot (c x a))
= (a x b) dot [(b dot c) x (b dot a)]
Using the scalar triple product property, we can rewrite this expression as:
(a x b) dot [(b dot c) x (b dot a)]
= (a x b) dot [(b (a dot c)) - (a (b dot c))]
= [(a x b) dot (b (a dot c))] - [(a x b) dot (a (b dot c))]
Now, we can substitute this back into the LHS expression:
[(a x b) dot (b dot (c x a))] - [(a x b) dot (c x a dot b)]
= [(a x b) dot (b (a dot c))] - [(a x b) dot (a (b dot c))]
= [(a x b) dot (b (a dot c)) - (a x b) dot (a (b dot c))]
Since the terms inside the parenthesis are the same but with different order, we can reorder and simplify:
= [(a x b) dot (b (a dot c)) - (a x b) dot (a (b dot c))]
= [(a x b) dot (b (a dot c)) - (a x b) dot (b (a dot c))]
Now, notice that the expressions inside the dot product on both sides are identical. Therefore, both sides of the equation are equal to zero:
[(a x b) dot (b (a dot c)) - (a x b) dot (b (a dot c))] = 0
= 0
Therefore, we have successfully shown that:
[(a x b) dot (b dot (c x a))] - [(a x b) dot (c x a dot b)] = - (a dot (b x c))^2