12.5 Surfaces in 3D

Lines in 3D

• Line equation

  • A line passing through the point $(x_0,y_0,z_0)$ and parallel to the vector $v=(a,b,c):$

    • \begin{align}x=x_{0}+at\\y=y_{0}+bt\\z=z_{0}+ct\end{align}
      • t is any real number
      • this is a parametric equation because this is what must be done

Planes in 3D

• Geometry of a plane

  • a plane can be said as a perpendicular direction and a point

  • OR three points are not co-linear

    • like the funny windmill

• Find a vector that is perp to a plane

  • A plane can be described as 3 points

    • then we find the vector to each of the points $\vec{a}=P_1-P_2,\vec{b}=P_1-P_3$ then we find the cross prod of the resultant vectors

• Fact

  • if $\theta$ is the angle between $\vec{a}\text{ and }\vec{b}$ then

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

  • two nonzero vectors are parallel if the cross product is the 0 vector

  • We can find the area of a parallelogram by taking the cross product of the two vectors

• Equations of planes

  • Vector equation

    • point P with the position vector ${\vec{r}}_0$ and normal direction $\vec{n}$:

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

  • Scalar equation:

    • Point $P(x_0,y_0,z_0)$ and normal vector $\vec{n}=(a,b,c)$:

      • $a(x-x_0)+b(y-y_0)+c(z-z_0)=0$
      • $ax+by+cz=d,d=a{x}_0+x{y}_0+c{z}_0$

• Distance between points and planes

  • in order to find the distance D from point $P_1(x_1,y_1,z_1)$ to the plane $ax+by+cz+d=0$, we let $P_0(x_0,y_0,z_0)$ be any point d be the vector corresponding to $\overrightarrow{P_0{P}_1}$

    • we then given plane ancan see that the distance is the projection of the point vector projected onto the next

Example

• Use the scalar triple product to determine whether poiints $P(1,0,1),Q(2,4,6),R(3,-1,2),S(6,2,8)$ lie in the same plane

  • We can take the scalar product of the 3 vectors (PQ,PR,PS) and if it is 0 then it is in the same plane

    • (this is because the triple product is finding the volume of the vectors)

$$\begin{array}{r}\vec{c}\cdot (\vec{b}\times \vec{a})=\left[\begin{array}{ccc}{c}_1& c_2& c_3\\ b_1& b_2& b_3\\ a_1& a_2& a_3\end{array}\right]\\ \text{we can take the triple product where c=PQ, b=PR, a=PS}\\ \left[\begin{array}{ccc}1& 4& 5\\ 2& -1& 1\\ 5& 2& 7\end{array}\right]⟹0\\ \text{ therforethe points lie in the same plane}\end{array}$$