Let a=6 i+9 j+12 k, b= i+11 j-2 k, and c satisfy a( c- b)= 0.

Question

Let $\vec a=6\hat i+9\hat j+12\hat k$, $\vec b=\alpha\hat i+11\hat j-2\hat k$, and $\vec c$ satisfy $\vec a	imes(\vec c-\vec b)=\vec 0$. If $\vec a\cdot\vec c=-12$ and $\vec c\cdot(\hat i-2\hat j+\hat k)=5$, then $\vec c\cdot(\hat i+\hat j+\hat k)$ is \_\_\_\_\_\_\_.

Answer: 11

Accepted Answer

$\vec c-\vec b\parallel\vec a \Rightarrow \vec c=\vec b+\lambda\vec a$ for some scalar; equivalently the source solves the resulting linear system in components $(x,y,z)$ of $\vec c$ directly from the three given scalar conditions along with the parallel condition, obtaining $\vec c=(23,2,-14)$.

$$\vec c\cdot(\hat i+\hat j+\hat k) = 23+2-14 = 11$$

Solutions · 1