Find an equation of a plane π that passes through point P = (0, 1, 0) and is parallel to vectors ~a = [−1, 3, 0], ~b = [3, 1, −5]

Question

I need some help with the second one, can someone help me to solve this?enter image description here

Answer 1

Hint: Write $$\vec{x}=\vec{OP}+s\vec{a}+t\vec{b}$$ where $s,t$ are real numbers. It is $$\vec{x}=[0,1,0]+s[-1,3,0]+t[3,1,-5]$$ and from here we get the system $$x=-s+3t$$ $$y=1+3s+t$$ $$z=-5t$$ It is better for you now? With the equation $$t=-\frac{z}{5}$$ you can eliminate $t$ then with $$s=-x-\frac{3}{5}z$$ you can eliminate $s$