• Geometric Vector
• vector
  • vector addition
  • scalar multiplication
  • component of vector
  • standard basis vector
  • dot product (scalar product, inner product)
    • angle between $\vec{a},\vec{b}$, $\theta$
      • perpendicular vector
    • direction angle
    • projection
  • cross product
    • magnitude of cross product
    • parallel vector
    • property of cross product
    • triple product
      • scalar triple product
      • vector triple product
• line $L$
• plane
  • plane-plane relation
  • point-plane distance
• cylinder
  • quadratic surface
    • standard form of quadratic surface
• vector function (vector-valued function)
  • limit of vector function
    • continuity of vector function
  • space curve
  • derivative of vector function
    • unit tangent vector
  • integral of vector function
  • arc length
  • curvature
    • smooth parametrized curve
  • (principal) unit normal vector
    • osculating plane
      • osculating circle (circle of curvature)
  • binormal vector
    • normal plane
  • velocity vector
    • speed (magnitude of velocity vector)
  • acceleration vector

Geometric Vector

vector

$$\vec{v}=\overrightarrow{AB}$$

• $\vec{v}$ displacement vector
• $A$ initial point
• $B$ terminal point
• $\vec{0}$ zero vector
• $\frac{\vec{v}}{|\vec{v}|}$ unit vector

vector addition

• triangle law
• parallelogram law

scalar multiplication

$$c\vec{v}$$

component of vector

$$\vec{a}=⟨a_1,a_2,a_3⟩$$

standard basis vector

$$\vec{j}=⟨1,0,0⟩\vec{k}=⟨0,1,0⟩\vec{l}=⟨0,0,1⟩$$

dot product (scalar product, inner product)

$$\begin{gathered} \vec{a}\cdot \vec{b}=\sum _i{a}_i{b}_i \\ =|\vec{a}||\vec{b}|\cos \theta \end{gathered}$$

angle between $\vec{a},\vec{b}$, $\theta$

$$\cos \theta =\frac{\vec{a}\cdot \vec{b}}{|\vec{a}||\vec{b}|}$$

perpendicular vector

$$\vec{a}\perp \vec{b}\iff \vec{a}\cdot \vec{b}=0$$

direction angle

$$\cos \alpha =\frac{a_1}{|\vec{a}|}\cos \beta =\frac{a_2}{|\vec{a}|}\cos \gamma =\frac{a_3}{|\vec{a}|}$$

• relation

  $${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$$

• conclusion

  $$\frac{\vec{a}}{|\vec{a}|}=⟨\cos \alpha ,\cos \beta ,\cos \gamma ⟩$$

projection

$$\begin{gathered} {\text{comp}}_{\vec{a}}\vec{b}=\frac{\vec{a}\cdot \vec{b}}{|\vec{a}|} \\ {\text{proj}}_{\vec{a}}\vec{b}=\frac{\vec{a}\cdot \vec{b}}{|\vec{a}{|}^2}\vec{a} \end{gathered}$$

cross product

$$\begin{gathered} \vec{a}\times \vec{b}=⟨a_2{b}_3-a_3{b}_2,a_3{b}_1-a_1{b}_3,a_1{b}_2-a_2{b}_1⟩ \\ =\left|\begin{array}{ccc}\vec{i}& \vec{j}& \vec{k}\\ a_1& a_2& a_3\\ b_1& b_2& b_3\end{array}\right| \end{gathered}$$

• $\vec{a}\times \vec{b}\perp \vec{a}$
• $\vec{a}\times \vec{b}\perp \vec{b}$
• right hand rule determine direction of $\vec{a}\times \vec{b}$

magnitude of cross product

$$|\vec{a}\times \vec{b}|=|\vec{a}||\vec{b}|\sin \theta$$

• equal to area of parallelogram by $\vec{a},\vec{b}$

parallel vector

$$\vec{a}\parallel \vec{b}\iff \vec{a}\times \vec{b}=\vec{0}$$

property of cross product

• $\vec{a}\times \vec{b}=-\vec{b}\times \vec{a}$
• linear
• $\vec{a}\cdot (\vec{b}\times \vec{c})=(\vec{a}\times \vec{b})\cdot \vec{c}$
• $\vec{a}\times (\vec{b}\times \vec{c})=(\vec{a}\cdot \vec{c})b-(\vec{a}\cdot \vec{b})\vec{c}$

triple product

scalar triple product

$$\vec{a}\cdot (\vec{b}\times \vec{c})$$

• volume of parallelepiped by $\vec{a},\vec{b},\vec{c}$

  $$V=|\vec{a}\cdot (\vec{b}\times \vec{c})|$$

• $V=0\iff \vec{a},\vec{b},\vec{c}$ coplanar

vector triple product

$$\vec{a}\times (\vec{b}\times \vec{c})$$

line $L$

• vector equation

  $$\vec{r}={\vec{r}}_0+t\vec{v}$$

• parametric equation

  $$x=x_0+aty=y_0+btz=z_0+ct$$

  where

  $$\vec{r}=⟨x,y,z⟩{\vec{r}}_0=⟨x_0,y_0,z_0⟩\vec{v}=⟨a,b,c⟩$$

• symmetric equation

  $$\frac{x-x_0}{a}=\frac{y-y_0}{b}=\frac{z-z_0}{c}$$

plane

• vector equation

  $$\vec{n}\cdot (\vec{r}-{\vec{r}}_0)=0$$

  • $\vec{n}$ normal vector

• scalar equation

  $$a(x-x_0)+b(y-y_0)+c(z-z_0)=0$$

  where

  $$\vec{n}=⟨a,b,c⟩$$

• linear equation

  $$ax+by+cz+d=0$$

plane-plane relation

• parallel plane
  normal vector parallel
• angle between plane
  acute angle between normal vector

point-plane distance

$$\begin{gathered} D=|{\text{comp}}_{\vec{n}}\vec{b}| \\ =\frac{|a{x}_1+b{y}_1+c{z}_1+d|}{\sqrt{a^2+b^2+c^2}} \end{gathered}$$

cylinder

consist of parallel ruling through a plane curve

quadratic surface

second degree equation in three variable

standard form of quadratic surface

obtained after translation and rotation

$$A{x}^2+B{y}^2+C{z}^2+J=0$$

or

$$A{x}^2+B{y}^2+Iz=0$$

• ellipsoid
  all traces are ellipses
• cone
  horizontal traces are ellipses vertical traces in the planes $x=k$ and $y=k$ are hyperbolas if $k\ne 0$ but are pairs of lines if $k=0$.
• elliptic paraboloid
  horizontal traces are ellipses
  vertical traces are parabolas
  variable to the first power indicate axis of paraboloid
• hyperboloid of one sheet
  horizontal traces are ellipses
  vertical traces are hyperbolas
  axis of symmetry corresponds to variable with negative coefficient
• hyperbolic paraboloid
  horizontal traces are hyperbolas
  vertical traces are parabolas
• hyperboloid of two sheets
  horizontal traces in $z=k$ are ellipses if $k>c$ or $k<-c$ vertical traces are hyperbolas two minus signs indicate two sheets.

vector function (vector-valued function)

$$\vec{r}(t)=⟨f(t),g(t),h(t)⟩$$

where $f(t),g(t),h(t)$ are component function of $\vec{r}$

limit of vector function

take limit of its component function

continuity of vector function

$\vec{r}$ continuous at $a\iff$

$$\underset{t\to a}{\lim }\vec{r}(t)=\vec{r}(a)$$

space curve

set $C$ of all point

$$x=f(t)y=g(t)z=h(t)$$

• parametric equation
• $t$ parameter
• visualization
  draw projection onto three plane

derivative of vector function

tangent vector

$$\begin{gathered} \frac{d\vec{r}}{dt}={\vec{r}}^{\prime }(t)=\underset{h\to 0}{\lim }\frac{\vec{r}(t+h)-\vec{r}(t)}{h} \\ =⟨f^{\prime }(t),g^{\prime }(t),h^{\prime }(t)⟩ \end{gathered}$$

• all derivative rule apply except quotient rule
  • dot product and cross product have separate rule
  • inner part of chain rule must be scalar function
• $|\vec{r}(t)|=C$
  $\Rightarrow \frac{d}{dt}(\vec{r}(t)\cdot \vec{r}(t))=0\Rightarrow \vec{r}(t)\cdot {\vec{r}}^{\prime }(t)=0$

unit tangent vector

$$\vec{T}(t)=\frac{{\vec{r}}^{\prime }(t)}{|{\vec{r}}^{\prime }(t)|}$$

integral of vector function

integrate each component function

arc length

$$\begin{gathered} s(t)={\int }_a^t\frac{ds}{du}du \\ ={\int }_a^t|{\vec{r}}^{\prime }(u)|du \end{gathered}$$

curvature

$$\begin{gathered} \kappa =\left|\frac{d\vec{T}}{ds}\right| \\ =\frac{|{\vec{T}}^{\prime }|}{|{\vec{r}}^{\prime }|} \\ =\frac{|{\vec{r}}^{\prime }\times {\vec{r}}^{\prime \prime }|}{|{\vec{r}}^{\prime }{|}^3} \end{gathered}$$

smooth parametrized curve

${\vec{r}}^{\prime }$ continuous and ${\vec{r}}^{\prime }\ne \vec{0}$

(principal) unit normal vector

$$\vec{N}(t)=\frac{{\vec{T}}^{\prime }(t)}{|{\vec{T}}^{\prime }(t)|}$$

• $\vec{N}(t)\perp \vec{T}(t)$
• $\vec{N}(t)\perp {\vec{r}}^{\prime }(t)$

osculating plane

determined by $\vec{T},\vec{N}$

osculating circle (circle of curvature)

• in osculating plane

• tangent the curve

• center towards $\vec{N}$

• radius

  $$\rho =\frac{1}{\kappa }$$

binormal vector

$$\vec{B}(t)=\vec{T}(t)\times \vec{N}(t)$$

• $\vec{B}(t)\perp \vec{T}(t)$
• $\vec{B}(t)\perp \vec{N}(t)$

normal plane

determined by $\vec{N},\vec{B}$

velocity vector

$$\vec{v}={\vec{r}}^{\prime }$$

speed (magnitude of velocity vector)

$$v=|\vec{v}(t)|=|{\vec{r}}^{\prime }(t)|=\frac{ds}{dt}$$

• $\vec{v}=v\vec{T}$
• $|{\vec{T}}^{\prime }|=\kappa v$

acceleration vector

$$\begin{gathered} \vec{a}={\vec{v}}^{\prime }={\vec{r}}^{\prime \prime } \\ =v^{\prime }\vec{T}+v{\vec{T}}^{\prime } \\ =v^{\prime }\vec{T}+\kappa v^2\vec{N} \\ =a_T\vec{T}+a_N\vec{N} \\ =\frac{{\vec{r}}^{\prime }(t)\cdot {\vec{r}}^{\prime \prime }(t)}{|{\vec{r}}^{\prime }(t)|}\vec{T}+\frac{|{\vec{r}}^{\prime }(t)\times {\vec{r}}^{\prime \prime }(t)|}{|{\vec{r}}^{\prime }(t)|}\vec{N} \end{gathered}$$