Vectors
Reciprocal system of vectors — identities
MJAT_TS2_P2
Grade 12
Question:
Let $\{\vec{a},\vec{b},\vec{c}\}$ be a set of three non-coplanar vectors and $\{\vec{p},\vec{q},\vec{r}\}$ be their reciprocal system: $\vec{p}=\dfrac{\vec{b}\times\vec{c}}{[\vec{a}\vec{b}\vec{c}]}$, $\vec{q}=\dfrac{\vec{c}\times\vec{a}}{[\vec{a}\vec{b}\vec{c}]}$, $\vec{r}=\dfrac{\vec{a}\times\vec{b}}{[\vec{a}\vec{b}\vec{c}]}$. Let $[\vec{a}\vec{b}\vec{c}]=V\neq 0$. Which of the following statements are correct?
A) $\vec{p}\times\vec{q}+\vec{q}\times\vec{r}+\vec{r}\times\vec{p} = \dfrac{\vec{a}+\vec{b}+\vec{c}}{V}$
B) $[(\vec{a}\times\vec{b})\times(\vec{b}\times\vec{c})]\cdot\vec{p} = V(\vec{b}\cdot\vec{p})$
C) $\vec{a}\times(\vec{b}\times\vec{c})+\vec{b}\times(\vec{c}\times\vec{a})+\vec{c}\times(\vec{a}\times\vec{b}) = V(\vec{p}+\vec{q}+\vec{r})$
D) $\vec{a}\cdot\vec{p}+\vec{b}\cdot\vec{q}+\vec{c}\cdot\vec{r}=3$
Step-by-Step Solution
Key Concept: Key properties of reciprocal system: $\vec{a}\cdot\vec{p}=\vec{b}\cdot\vec{q}=\vec{c}\cdot\vec{r}=1$ and $\vec{a}\cdot\vec{q}=\vec{a}\cdot\vec{r}=0$, etc. So D: $\vec{a}\cdot\vec{p}+\vec{b}\cdot\vec{q}+\vec{c}\cdot\vec{r}=1+1+1=3$ ✓.
A ✓ (cross product identity for reciprocal system). B ✓ (scalar triple product). C ✗ (incorrect identity). D ✓. Answer: A, B, D.
Correct Answer: ABD