Vectors
Cauchy-Schwarz equality and ratio computation
MJAT_TS3_P2
Grade 12
Question:
If $a,b,c,p,q,r\in\mathbb{R}\setminus\{0\}$ such that $ap+bq+cr+\sqrt{(a^2+b^2+c^2)(p^2+q^2+r^2)}=0$, then find the value of $\dfrac{bp}{aq}+\dfrac{cq}{br}+\dfrac{cp}{ar}$.
Step-by-Step Solution
Key Concept: The condition $ap+bq+cr = -\sqrt{(a^2+b^2+c^2)(p^2+q^2+r^2)}$ means $|ap+bq+cr|=\sqrt{(a^2+b^2+c^2)(p^2+q^2+r^2)}$. This is equality in Cauchy-Schwarz with the dot product being negative: vectors $(a,b,c)$ and $(p,q,r)$ are antiparallel.
$(p,q,r)=-k(a,b,c)\Rightarrow$ each ratio $=1$. Sum $=\mathbf{3}$.
Correct Answer: 3