Find the piercing point of vec r×(4vec i-3vec j+vec k)=-6vec…

Find the piercing point of $\vec r\times(4\vec i-3\vec j+\vec k)=-6\vec i-8\vec j$ through $\vec r\cdot(3\vec i+5\vec j-\vec k)=2$.

Step 1: Solution

Here $\vec p=\{4,-3,1\}$, $\vec q=\{-6,-8,0\}$, $\vec N=\{3,5,-1\}$, $\alpha=2$: \[\vec p\cdot\vec N=-4,\qquad \vec N\times\vec q=\{-8,6,6\},\qquad \alpha\vec p+\vec N\times\vec q=\{0,0,8\}.\]

\[\vec r=\frac{\{0,0,8\}}{-4}:\qquad P(0,0,-2)\]

Check: $\vec r\cdot\vec N=2$, and $\{0,0,-2\}\times\{4,-3,1\}=\{-6,-8,0\}$.