Answer
$8x+y-2z=31$
Work Step by Step
To find the Normal vector in order to find the equation of plane we need to find two direction vectors and get their cross-product.
The first direction vector is obtained from the equation of the line given to us. and we can find second by finding vector from point x1,y1,z1(3,5,-1) and second from the equation of line x2,y2,z2(4,-1,0). The second direction vector comes out to be $\lt1,-6,1\gt$ [x2-x1,y2-y1,z2-z1]
The cross product of both dir. vectors gives $\lt8,1,-2\gt$
The general form of the equation of the plane passing through the point $(a,b,c)$ and having normal vector $\lt l,m,n\gt$is:
$l(x-a)+m(y-b)+n(z-c)=0$
Thus, the equation of plane is:
$-16(x-3)-2(y-5)+4(z+1)=0$
After simplification, we get
$-16x-2y+4z+62=0$
Or, $8x+y-2z=31$
