Equation of dot product and cross product

Question

I know basic concept of dot product and cross product.
but I'm not sure how to solve the question below or where to look for.

I need to find the choice that is not correct.

Answer 1

(1) is a special case of Binet-Cauchy identity described HERE

(2) has the opposite sign on the right side so it's definitely wrong.

For (3), let's start with the formula for the triple product proved HERE:

$$\mathbf{a}\times(\mathbf{b}\times\mathbf{c})=\mathbf{b}(\mathbf{a}\cdot\mathbf{c})-\mathbf{c}(\mathbf{a}\cdot\mathbf{b})$$

Swap places of operands on the left and the whole expression changes sign:

$$(\mathbf{b}\times\mathbf{c})\times\mathbf{a}=\mathbf{c}(\mathbf{a}\cdot\mathbf{b})-\mathbf{b}(\mathbf{a}\cdot\mathbf{c})$$

Now replace $\mathbf{b}$ with $\mathbf{a}$, $\mathbf{c}$ with $\mathbf{b}$ and $\mathbf{a}$ with $\mathbf{c}\times\mathbf{d}$ and you get:

$$(\mathbf{a}\times\mathbf{b})\times(\mathbf{c}\times\mathbf{d})=\mathbf{b}((\mathbf{c}\times\mathbf{d})\cdot\mathbf{a})-\mathbf{a}((\mathbf{c}\times\mathbf{d})\cdot\mathbf{b})$$

...which is exactly (3).

Answer 2

HINT

The first one is the Binet–Cauchy identity.