# Plotting a paraboloid and a plane

I am trying to visualize the intersection of a paraboloid and and a plane. My paraboloid is of the form $$f(x) = \frac{1}{2} x^\top A x - b^\top x$$ where $$x$$ is a vector and $$A$$ is the pd matrix $$A = \begin{bmatrix} 3 & 2\\ 2 & 6 \end{bmatrix}$$ and $$b = \begin{bmatrix} 2 \\ -8 \end{bmatrix}$$ My plane is parameterised as $$x^{t+1} = x^t + \alpha (b -Ax^t)$$

Basically my plane corresponds to a line search to choose $$\alpha$$ to minimize $$f$$ along a line.

I plotted the function $$f$$ using Plot3D, but I do not know how to plot the plane and contours.

My code so far is just plotting the function

Plot3D[1.5 x^2 + 2  x y + 3 y^2 + 2 x - 8 y, {x, -4,4}, {y, -4,4}]


I am not sure how to plot the plane corresponding to

$$x^{t+1} = x^t + \alpha (b -Ax^t)$$

• What did you do? show your code. – Alex Trounev Apr 20 at 16:55
• @AlexTrounev. i edited the question. – Shew Apr 21 at 19:45
• There is no plane equation. Usually in geometry used $ax+by+cz+d=0$. In Mathematica used InfinitePlane[{p1,p2,p3}]  or InfinitePlane[{p1, {v1,v2}}] . – Alex Trounev Apr 21 at 23:55