MAT223 Lecture 12

• 1
  • Let $\overrightarrow{x}=\left[\begin{array}{c}1\\ 3\\ 2\end{array}\right]$ and $\overrightarrow{y}=\left[\begin{array}{c}2\\ 1\\ 1\end{array}\right]$
  • $\overrightarrow{x}\cdot \overrightarrow{y}=(1)(2)+(3)(1)+(2)(1)=7$
• 2
  • Which vectors are $\perp$ to both $\overrightarrow{x}=\left[\begin{array}{c}1\\ 1\\ 2\end{array}\right]$ and $\overrightarrow{y}=\left[\begin{array}{c}1\\ -1\\ 1\end{array}\right]$
  • This is the cross product
  • $\overrightarrow{x}\times \overrightarrow{y}=\left|\begin{array}{ccc}\overrightarrow{i}& \overrightarrow{j}& \overrightarrow{k}\\ 1& 1& 2\\ 1& -1& 1\end{array}\right|$
  • $\overrightarrow{x}\times \overrightarrow{y}=⟨[(1)(1)-(2)(-1)],[(1)(1)-(2)(1)],[(1)(-1)-(1)(1)]⟩$
  • $\overrightarrow{x}\times \overrightarrow{y}=⟨3,-1,-2⟩$
• 4
  • If $L$ goes through $P$ and $Q$
  • What's the direction vector?
  • $\overrightarrow{PQ}$ or $\overrightarrow{QP}$
  • What about $P_{0}P$ where $P_0$ is any point on $L$?
    • $\overrightarrow{P_{0}P}$ is parallel to $\overrightarrow{PQ}$ and $\overrightarrow{QP}$
• 6
  • Match each plane with it's normal vector:
    • $x-y+z=2$
      • $⟨1,-1,1⟩$
    • $y+z=x+3$
      • $-x+y+z=3$
      • $⟨-1,1,1⟩$
    • $3x-4y-z=5$
      • $⟨3,-4,-1⟩$
    • $x-z=4$
      • $x-0y-z=4$
      • $⟨1,0,-1⟩$
• Activity 1
  • Find equations for planes and lines:
  • Two lines intersections at a single point:
    • $L_1:⟨0,0,1⟩+t⟨0,1,1⟩$
    • $L_2:⟨0,0,1⟩+t⟨1,1,0⟩$
  • Two planes which intersect in a line:
    • $P_1:4(x-1)+2(y-1)+1(z-1)=0$
    • $P_2:1(x-1)+2(y-1)+8(z-1)=0$
    • Might be easier to just shift a plane (like $yz$) plane by $1$.
    • $P_1:1(x-1)+0(y-0)+0(z-0)=0$
    • $P_2:0(x-0)+1(y-1)+0(z-0)=0$
    • Their intersection is the line:
    • $L:⟨1,1,0⟩+t⟨0,0,1⟩$
    • Or we can do this with linear algebra:
    • A system with 3 variables and 2 equations will have infinitely many solutions, which is a line. Exactly one parameter.
    • $\left[\begin{array}{cccc}1& 0& 1& 1\\ 0& 1& 1& 2\end{array}\right]$
    • Find the intersection:
      • $\overrightarrow{x}=\left[\begin{array}{c}1-x_3\\ 2-x_3\\ 1\end{array}\right]$
      • $\overrightarrow{x}=\left[\begin{array}{c}1-z\\ 2-z\\ t\end{array}\right]$
      • $\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}1\\ 2\\ 0\end{array}\right]+t\left[\begin{array}{c}-1\\ -1\\ 1\end{array}\right]$
      • $L:⟨1,2,0⟩+t⟨-1,-1,1⟩$
  • Two planes with no intersection:
    • $P_1:(x-1)+(y-1)+(z-1)=0$
    • $P_1:(x-1)+(y-1)+(z-2)=0$
  • Three planes which intersect in a line:
    • $P_1:1(x-1)+1(y-1)+1(z-1)=0$
    • $P_2:2(x-1)+3(y-1)+1(z-1)=0$
    • $P_3:1(x-1)+1(y-1)+0(z-1)=0$
  • Three planes which intersect at a single point:
    • Shifting the $xy,yz,xz$ planes.
    • $P_1:1(x-1)+0(y-0)+0(z-0)=0$
    • $P_2:0(x-0)+1(y-2)+0(z-0)=0$
    • $P_3:0(x-0)+0(y-0)+1(z-3)=0$
    • Linear algebra:
    • $\left[\begin{array}{cccc}1& 0& 1& 0\\ 0& 1& 0& 1\\ 0& 0& 1& 0\end{array}\right]$
    • $\overrightarrow{x}=\left[\begin{array}{c}-t\\ 1\\ t\end{array}\right]$
    • $\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]+t\left[\begin{array}{c}-1\\ 0\\ 1\end{array}\right]$
  • Two lines so that no plane contains both:
    • Two skew lines.
    • $P_1:1(x-1)+1(y-1)+1(z-1)=0$
    • One line in $P_1$:
      • $L_1:⟨1,1,1⟩+t⟨1,1,1⟩$
    • One line not in $P_1$:
      • $L_1:⟨1,1,8⟩+t⟨1,1,1⟩$
• 7
  • $\overrightarrow{v}=\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$
  • $\overrightarrow{w}=\left[\begin{array}{c}-2\\ 1\\ 1\end{array}\right]$
  • ${\mathrm{proj}}_{\overrightarrow{w}}\overrightarrow{v}=\frac{\overrightarrow{v}\cdot \overrightarrow{w}}{{\left|\overrightarrow{w}\right|}^2}\overrightarrow{w}$
  • ${\mathrm{proj}}_{\overrightarrow{w}}\overrightarrow{v}=\frac{(1)(-2)+(2)(1)+(3)(1)}{(-2{)}^2+1^2+1^2}\left[\begin{array}{c}-2\\ 1\\ 1\end{array}\right]$
  • $\frac{(1)(-2)+(2)(1)+(3)(1)}{(-2{)}^2+1^2+1^2}=\frac{1}{2}$
  • ${\mathrm{proj}}_{\overrightarrow{w}}\overrightarrow{v}=\frac{1}{2}\left[\begin{array}{c}-2\\ 1\\ 1\end{array}\right]$
  • ${\mathrm{proj}}_{\overrightarrow{w}}\overrightarrow{v}=\left[\begin{array}{c}-1\\ \frac{1}{2}\\ \frac{1}{2}\end{array}\right]$
• Activity 2
  • Point to plan
  • Determine the distance from $P(2,1,-3)$ to $3x-y+4z=1$ as well as point $Q(x,y,z)$ on the plane to which $P$ is closest.
  • $\overrightarrow{n}=⟨3,-1,4⟩$
  • $P_0(0,-1,0)$ is on the plane?
    • $(3)(0)-(1)(-1)+(4)(0)=1$
    • So it is.
  • $\overrightarrow{u}=\overrightarrow{P_{0}P}=⟨2,2,-3⟩$
  • ${\overrightarrow{u}}_1=\overrightarrow{QP}=⟨2-x,1-y,-3-z⟩$
  • ${\overrightarrow{u}}_1$ is ${\mathrm{proj}}_{\overrightarrow{n}}\overrightarrow{u}=(-\frac{4}{13})\left[\begin{array}{c}3\\ -1\\ 4\end{array}\right]$
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