Question

If $u,v,w$ are non-coplanar vector and $p,q$ are real numbers, then the equality $\left[3upvpw\right]-\left[pvwqw\right]-\left[2wqvqu\right]=0$ holds for

• A. Exactly two values of $(p,q)$
• B. More than but not all values of $(p,q)$
• C. All values of $(p,q)$
• D. Exactly one values of $(p,q)$

Answer 1

The correct option is A Exactly two values of $(p,q)$

$\begin{array}{c}Since,\\ \left[3upvpw\right]-\left[pvwqqv\right]=0\\ \therefore 3p^2\left[u.\left(v\times w\right)\right]-pq\left[v.\left(w\times u\right)\right]\\ -2q^2\left[w.\left(v\times u\right)\right]=0\\ \Rightarrow \left(3p^2-pq+2q^2\right)\left[u.\left(v\times w\right)\right]=0\\ But\left[uvw\right]\ne 0\\ \Rightarrow 3p^2-pq+2q^2=0\\ \therefore p=q=0\end{array}$