Practicing Success
If $\vec a,\vec b,\vec c$ are non-coplanar vectors and λ is a real number, then $\begin{bmatrix}λ(\vec a+\vec b) &λ^4\vec b& λ\vec c\end{bmatrix}=\begin{bmatrix}\vec a&\vec b+\vec c&\vec b\end{bmatrix}$ for |
exactly two values of λ exactly three values of λ no value of λ exactly one value of λ |
no value of λ |
We have, $\begin{bmatrix}λ(\vec a+\vec b) &λ^4\vec b& λ\vec c\end{bmatrix}=\begin{bmatrix}\vec a&\vec b+\vec c&\vec b\end{bmatrix}$ $⇒λ^4\begin{bmatrix}\vec a+\vec b&\vec b&\vec c\end{bmatrix}=\begin{bmatrix}\vec a&\vec b+\vec c&\vec b\end{bmatrix}$ $⇒λ^4\left\{[\vec a\,\,\vec b\,\,\vec c]+[\vec a\,\,\vec b\,\,\vec c]\right\}=\left\{[\vec a\,\,\vec b\,\,\vec b]+[\vec a\,\,\vec c\,\,\vec b]\right\}$ $⇒λ^4[\vec a\,\,\vec b\,\,\vec c]=-[\vec a\,\,\vec b\,\,\vec c]$ $⇒(λ^4+1)[\vec a\,\,\vec b\,\,\vec c]=0$ This is not true for any real value of λ as $[\vec a\,\,\vec b\,\,\vec c]≠0$. |