Year 12s: what will be the first way you research your future options? >>

Line normal to a scalar field

Say I have a scalar field

\phi = (x^3)y + (x^2)z +xy + 3

and I want to find the line tangent to the plane at point P(1,1,1) when

\phi = 6

. Would I have to do

\nabla \phi

at point (1,1,1) ?

That would give me

(3x^2 y + 2xz + y)\vec{i} + (x^3 + x)\vec{j} + (x^2)\vec{k}

and when I plug in the values I'd get 6i + 2j + k and so the equation of the line would be

\mathbf{r} = \begin{pmatrix} 1\\1\\1 \end{pmatrix} + \lambda \begin{pmatrix} 6\\2\\1 \end{pmatrix}

.

The problem I have with this is I was told the gradient points in the direction of steepest ascent, so I don't understand how it can represent the vector normal to the plane at that point and represent the steepest ascent at the point as both are different vectors...Can anyone help me clear my confusion?