Study guide and
4 practice problems
on:
Lines in 2d can be specified by a point and a normal vector
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We can specify a line in 2d by a point $\mathbf{x_0}$ and a normal vector $\mathbf{n}$.
A 2d line with normal vector $\bfn$ is given by $$\bfn\cdot \bfx = b$$ for some $b$.
The 2d line going through $\bfx_0$ with normal vector $\bfn$ is given by $$\bfn \cdot \bfx = \bfn \cdot \bfx_0$$
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Practice problems
Find the line containing the points $(0, 1)$ and $(3, 0)$ by computing a normal vector to the line. Write it in the form $\bfn \cdot \bfx = b$.
Solution
Find a normal vector to the 2d line $2x - y = 5$.
Solution
Sketch the region in 2d satisfied by $ ( 1, 1 ) \cdot \bfx \geq 2.$
Solution
Find a normal vector to the curve $\sqrt{x} + \sqrt{y} = 2$ at $(x,y) = (1,2)$. Use it to find the tangent line at $(1,1)$ expressed in the form $\bfn \cdot \bfx = b$.
Solution