Proof with vector properties $a = \frac{\vec{D}·(\vec{B} \times \vec{C} ) }{\vec{A} · (\vec{B} \times \vec{C} )}$

Question

I need to prove that given 3 non parallel vectors such that $\vec{D} = a \vec{A} + b \vec{B} + c\vec{C}$.

We can obtain $a = \frac{\vec{D}·(\vec{B} \times \vec{C} ) }{\vec{A} · (\vec{B} \times \vec{C} )}$

I don't even know where to start and hope you can give me some help. Thanks.

Answer 1

Using invariance of the triple product under circular shift one may see that $$ \vec{B}\cdot[\vec{B}\times\vec{C}] = \vec{C}\cdot[\vec{B}\times\vec{B}] = 0 \\ \vec{C}\cdot[\vec{B}\times\vec{C}] = \vec{B}\cdot[\vec{C}\times\vec{C}] = 0. $$ Thus

$$ \vec{D}\cdot[\vec{B}\times\vec{C}] = (a\vec{A} + b\vec{B} + c\vec{C})\cdot[\vec{B}\times\vec{C}] = a\vec{A}\cdot[\vec{B}\times\vec{C}], $$ which leads to the result required.