From the line equation you have:
- [tex]\vec{p}=(4,-1,-3)\parallel \text{ line};[/tex]
- [tex]B(-3,-3,-5)\in \text{ line.}[/tex]
About the plane you know that it should pass through the point [tex]A(-5,-1,2)[/tex] and be ortogonal to the given line (also ortogonal to the vector [tex]\vec{p}[/tex]). Then its equation is:
[tex]4\cdot (x-(-5))+(-1)\cdot (y-(-1))+(-3)\cdot (z-2)=0,\\ \\4(x+5)-(y+1)-3(z-2)=0,\\ \\4x+20-y-1-3z+6=0,\\ \\4x-y-3z+25=0.[/tex]
Answer: [tex]4x-y-3z+25=0.[/tex]
