Finding the Normal & Tangent Vector and write an Equation. Find a normal vector and a tangent vector at the
point P. Write an equation for the tangent line and an equation
for the normal line.

$x^2 + xy + y^2 = 3; P(−1,−1).$

So what I did first was find the gradient by taking the partial derivatives.

$▽f$ = $(2x+y)i + (x+2y)j$

$▽f$  $(-1,-1)$ = $3i-3j = 3$

Normal Vector $i+j$                   

Tangent Vector $i-j$

How does one find the tangent line and normal line from this?
Hint:

$\vec{r}$ represents the point (could be any points) on the line , and $\vec{x}$ is a known point on the line 

$\vec{r}-\vec{x}$ is the direction of the line.



P.S : I don't know why I earn a downvote . This is very basic knowledge in elementary geometry ( someone call it analytical geometry) . Therefore , I just provide some hints without details. Thanks to your imprudent judgement , I will keep silent.
Hint:

$\vec{r}$ represents the point (could be any points) on the line , and $\vec{x}$ is a known point on the line 

$\vec{r}-\vec{x}$ is the direction of the line.



P.S : I don't know why I earn a downvote . This is very basic knowledge in elementary geometry ( someone call it analytical geometry) . Therefore , I just provide some hints without details. Thanks to your imprudent judgement , I will keep silent.