Study guide and
4 practice problems
on:
A plane is specified by a point and a normal vector
$\newcommand{\bfA}{\mathbf{A}}$ $\newcommand{\bfB}{\mathbf{B}}$ $\newcommand{\bfC}{\mathbf{C}}$ $\newcommand{\bfF}{\mathbf{F}}$ $\newcommand{\bfI}{\mathbf{I}}$ $\newcommand{\bfa}{\mathbf{a}}$ $\newcommand{\bfb}{\mathbf{b}}$ $\newcommand{\bfc}{\mathbf{c}}$ $\newcommand{\bfd}{\mathbf{d}}$ $\newcommand{\bfe}{\mathbf{e}}$ $\newcommand{\bfi}{\mathbf{i}}$ $\newcommand{\bfj}{\mathbf{j}}$ $\newcommand{\bfk}{\mathbf{k}}$ $\newcommand{\bfn}{\mathbf{n}}$ $\newcommand{\bfr}{\mathbf{r}}$ $\newcommand{\bfu}{\mathbf{u}}$ $\newcommand{\bfv}{\mathbf{v}}$ $\newcommand{\bfw}{\mathbf{w}}$ $\newcommand{\bfx}{\mathbf{x}}$ $\newcommand{\bfy}{\mathbf{y}}$ $\newcommand{\bfz}{\mathbf{z}}$
To specify a plane, we need a point $\mathbf{x_0}$ and a normal vector $\mathbf{n}$.
A plane with normal vector $\bfn$ is given by $$\bfn\cdot \bfx = b$$ for some $b$.
The plane going through $\bfx_0$ with normal vector $\bfn$ is given by $$\bfn \cdot \bfx = \bfn \cdot \bfx_0.$$
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Practice problems
Consider the set of 3d points whose component (scalar projection) along the vector $\bfv$ is the constant $c$. Show that these points form a plane, and find that plane.
Solution
Find a normal vector to the plane $2x + y - 3z = 1$.
Solution
Find the plane containing the points $(a,0,0)$, $(0, b, 0)$, and $(0,0,c)$.
Solution
Find the tangent plane at $(1,1,1)$ to the surface $$x^2 + y^2 + z^2 + xy + xz = 5.$$
Solution
Give the equation for the tangent plane to the surface $z x^2 + x y^2 + y z^2 = 5$ at the point $(-1,1,2)$.
Solution