Gradient of a linear function

Consider the function $f : mathbf{R}^2 rightarrow mathbf{R}$ , with values

Its gradient is constant, with values

$nabla f = left(begin{array}{c} frac{partial f}{partial x_1}(x) frac{partial f}{partial x_2}(x) end{array} right) = left(begin{array}{c} 1 2 end{array} right).$

For a given t in mathbf{R} , the -level set is the set of points such that f(x) = t :

$mathbf{L}_t(f) := {(x_1, x_2) ~:~ x_1 + 2 x_2 = t }.$

The level sets are hyperplanes, and are orthogonal to the gradient.

The figure on the left shows that the gradient is orthogonal to the level sets.