MAT223 Lecture 13

• 1
  • $P=(2,1-3)$
  • Plane with $3x-y+4z=1$
  • $Q=(x,y,z)$ on the plane.
  • Find values for $Q$ which make it closest to $P$.
  • $L(t)=\overrightarrow{x}=t\overrightarrow{d}+{\overrightarrow{p}}_0$
  • Find $Q$ first.
  • I can pick an arbitrary point on the plane, say $P_0=(0,-1,0)$.
  • $\overrightarrow{d}=⟨3,1,-4⟩$ is the normal vector to the plane.
  • ${\overrightarrow{p}}_0=⟨0,-1,0⟩$ is the point on the plane.
  • $L(t)=⟨3t,1t-1,-4t⟩$
  • You can also do a line with point $Q$ and direction $\overrightarrow{n}$
    • $L(t)=⟨x,y,z⟩+t⟨3,1,-4⟩$
    • $L(t)=⟨x+3t,y+t,z-4t⟩$
  • Note $Q$ lies on line $L$ and plane. Then find a value of $t$ so $\overrightarrow{q}=\overrightarrow{p}+t\overrightarrow{n}$ . Then you find $Q$ by that.
  • Based on the line equation we have $\overrightarrow{n}\cdot \overrightarrow{q}=1$
  • $3x-y+4z=1=⟨2,1,-3⟩+t⟨3,-1,4⟩$
  • $3(2+3t)-(1-t)+4(-3+4t)=1$
  • $(3)(2+3t)-(1-t)+(4)(-3+4t)=26t-7$
  • $26t-7=1$
  • $t=\frac{4}{13}$
  • Real way:
    • Since $\overrightarrow{p}+t\overrightarrow{n}=\overrightarrow{q}$
    • $\overrightarrow{n}\cdot \overrightarrow{q}=1$
    • $\overrightarrow{n}(\overrightarrow{p}+t\overrightarrow{n})=1$
  • Solution:
    • $\overrightarrow{q}=\overrightarrow{p}+t\overrightarrow{n}$
    • $\overrightarrow{n}\cdot \overrightarrow{q}=\overrightarrow{n}\cdot (\overrightarrow{p}+t\overrightarrow{n})$
    • $1=\overrightarrow{n}\cdot \overrightarrow{p}+t(\overrightarrow{n}\cdot \overrightarrow{n})$
    • $1=\left[\begin{array}{c}3\\ -1\\ 4\end{array}\right]\cdot \left[\begin{array}{c}2\\ 1\\ -3\end{array}\right]+t(\left[\begin{array}{c}3\\ -1\\ 4\end{array}\right]\cdot \left[\begin{array}{c}3\\ -1\\ 4\end{array}\right])$
    • $1=-7+26t$
    • $t=\frac{4}{13}$
• Q1
  • $\overrightarrow{u}=\left[\begin{array}{c}1\\ -3\\ 2\end{array}\right]$
  • $\overrightarrow{v}=\left[\begin{array}{c}2\\ 0\\ -1\end{array}\right]$
  • $\overrightarrow{u}\times \overrightarrow{v}=\left|\begin{array}{ccc}\overrightarrow{i}& \overrightarrow{j}& \overrightarrow{k}\\ 1& -3& 2\\ 2& 0& -1\end{array}\right|$
  • $\overrightarrow{u}\times \overrightarrow{v}=\overrightarrow{i}(-1{)}^{1+1}((-3)(-1)-(2)(0))+\overrightarrow{j}(-1{)}^{1+2}((1)(-1)-(2)(2))+\overrightarrow{k}(-1{)}^{1+3}((1)(0)-(-3)(2))$
• Q2
  • $P=(1,2,3)$
  • $Q=(-1,1,-1)$
  • $R=(1,0,,1)$
  • Which are normal to the plane containing $P,Q,R$?
  • $\overrightarrow{PR}\times \overrightarrow{OP}$
    • Possibly not
  • $\overrightarrow{PQ}\times \overrightarrow{QR}$
    • Yes
  • $\overrightarrow{PQ}\times \overrightarrow{RP}$
    • Yes
  • $\overrightarrow{PQ}\times \overrightarrow{QP}$
    • No
• In class activity 1
  • $|\overrightarrow{v}\times \overrightarrow{w}|=\left|\overrightarrow{v}\right|\left|\overrightarrow{w}\right|\sin \theta$