MATH 251 Lecture 6
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Problem
Given A(1, 2, 3), B(4, 5, 6), and C(-5, -5, -5), find the equation of the plane through these points.
Find 2 vectors among the points
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \vec{u} &= \overrightarrow{B-A} = \left\langle 3,\, 3,\, 3 \right\rangle \\ \vec{v} &= \overrightarrow{C-A} = \left\langle -6,\, -7,\, -8 \right\rangle \end{align}}
Take their cross product: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \vec{n} = \vec{u} \times \vec{v} &= 3 \left( \hat\imath \left(-8+7 \right) - \hat\jmath \left( -8 + 6 \right) + \hat{k} \left( -7+6 \right) \right) \\ &= 3 \left\langle -1,\, 2,\, -1 \right\rangle \end{align}}
Therefore using normal form:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} -\left(x-1\right)+2\left(y-2\right)-\left(z-3\right)&=0 \\ -x+2y-z&=0 \end{align}}
Another Problem
Given the planes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x + 2y - 3z = 1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2x+3y-z=0} , find the distance between them.
The distance is 0 because they intersect: they have different normal vectors: (1, 2, -3) and (2, 3, -1)
Yet Another Problem
Given Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = x + 2y-3z = 1} andFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta = 2x + 4y - 6z = 9} , find the distance between the planes.
Both planes have the same normal vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{n} = \left\langle 1, 2, -3 \right\rangle} , so they are parallel.
Pick a point on plane Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} : P(1,0,0). Pick one from plane Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta} : Q(9/2,0,0).
The distance between the planes is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| \mathrm{comp}_{\vec{n}}\left(\overrightarrow{P-Q}\right) \right|}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathrm{comp}_{\vec{n}} \left( \overrightarrow{P-Q} \right) &= \frac{\left( \overrightarrow{P-Q} \right) \cdot \vec{n}}{\left\| \vec{n} \right\|} \\ &= \frac{3.5}{\sqrt{14}} \end{align}}
This guy has a lot of problems ;)
"line": Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha \left( t \right) = \begin{pmatrix}0\\-1\\2\end{pmatrix} + t \begin{pmatrix}1\\4\\-2\end{pmatrix}}
"plane": Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4x+2z=3}
Find the distance between these two things. We can determine whether they intersect by comparing the dot product of the plane's normal vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{n} = \left\langle 4,0,2 \right\langle} to the line's directional vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v} = \left\langle 1,4,-2 \right\rangle} .
Their dot product is 0, so they are indeed parallel.
Find a point on the line: P(0,-1,2).
Find a point on the plane: Q(0,0,3/2).
The distance between the plane and the point is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| \mathrm{comp}_{\vec{n}}\left(\overrightarrow{P-Q}\right) \right|}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathrm{comp}_{\vec{n}} \left( \overrightarrow{P-Q} \right) &= \left| \frac{\left( \overrightarrow{P-Q} \right) \cdot \vec{n}}{\left\|\vec{n}\right\|} \right| \\ &= \frac{1}{2\sqrt{5}} \end{align}}
Quadric Surfaces
Determined by a quadratic equation:
For example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2-2xy+z^2+3z-1=0} is a quadric surface.
For our purposes, we are not going to have terms like Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xy} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle yz} , or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xz} because adding those terms just rotates the surface. We will also omit linear terms of a single variable because that justs shifts the graph up and down.
Ellipsoid
A 3D oval...
Paraboloid
Start with a parabola and rotate it around the x axis
Hyperboloid of One Sheet
Start with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2-y^2=1} (a hyperbola) and rotate around the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} axis
Hyperboloid of Two Sheets
Start with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2-y^2=1} (a hyperbola) and rotate around the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} axis
Cylinder
hyperbolic, elliptic, parabolic
Omit Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} variable, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xy} plot sweeps out a "cylinder"