MATH 251 Lecture 2

« previous | Friday, January 20, 2012 | next »

Vectors and 3D Space

See MATH 152 Chapter 11.1, MATH 152 Chapter 11.2, and MATH 152 Chapter 11.3

Spheres

Given a center point $P$ and the radius $r$, the defined sphere is:

$S_{P}(r)=\left\{\mathrm {points} \ x~|~d(X,P)=r\right\}$
where $d(X,P)$ represents the distance between points $X$ and $P$.

And its (standard) equation is

$S_{P}(r)=\left(x-P_{x}\right)^{2}+\left(y-P_{y}\right)^{2}+\left(z-P_{z}\right)^{2}=r^{2}$

Example 1

Find the center and radius of the sphere with the equation $x^{2}+6x+y^{2}-4y+z^{2}+2z=0$

1. complete the square for all variables: $\left(x^{2}+6x+9\right)+\left(y^{2}-4y+4\right)+\left(z+2z+1\right)=0+9+4+1$
2. Factor: $(x+3)^{2}+(y-2)^{2}+(z+1)^{2}=14$
3. The center is at $\left(-3,\,2,\,-1\right)$, and the radius is ${\sqrt {14}}$

Dot Product

See MATH 152 Chapter 11.2#Dot Product and MATH 152 Chapter 11.2#3D Vectors

Prof's notation: ${\vec {v}}^{2}={\vec {v}}\cdot {\vec {v}}=v_{x}^{2}+v_{y}^{2}+v_{z}^{2}=\left|{\vec {v}}\right|^{2}$

Dot product gives distances via

$d(A,B)^{2}=\left({\overrightarrow {B-A}}\right)\cdot \left({\overrightarrow {B-A}}\right)$

ijk Notation

• ${\hat {\imath }}$ unit vector in $x$ direction
• ${\hat {\jmath }}$ unit vector in $y$ direction
• ${\hat {k}}$ unit vector in $z$ direction

Angle Between Vectors

$\theta =\arccos {\left({\frac {{\vec {v}}\cdot {\vec {w}}}{\left|{\vec {v}}\right|\left|{\vec {w}}\right|}}\right)}$