If a= i+ j+4 k, b=-2 i+4 j-2 k, and c=2 i+3 j+ k are coplanar, and the projection of a on …

Question

If $\vec a=\lambda\hat i+\mu\hat j+4\hat k$, $\vec b=-2\hat i+4\hat j-2\hat k$, and $\vec c=2\hat i+3\hat j+\hat k$ are coplanar, and the projection of $\vec a$ on $\vec b$ is $\sqrt{54}$ units, then the sum of all possible values of $\lambda+\mu$ is:

Accepted Answer

Coplanarity: $\begin{vmatrix}\lambda&\mu&4\-2&4&-2\2&3&1\end{vmatrix}=0 \implies \lambda(10)-\mu(2)+4(-14)=0 \implies 5\lambda-\mu=28$ \ldots  (1)

Projection: $\dfrac{\vec a\cdot\vec b}{|\vec b|} = \pm\sqrt{54}$, i.e. $\dfrac{-2\lambda+4\mu-8}{2\sqrt6}=\pm\sqrt{54}$, giving $-2\lambda+4\mu-8=\pm36$ \ldots  (2)

Solving (1) and (2) for each sign and summing the resulting $\lambda+\mu$ values (source's stated result):

Answer: 24

Solutions · 1