Lecture 03   

Lines & Planes

APMA E2000

Drew Youngren dcy2@columbia.edu

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$\gdef\RR{\mathbb{R}}$ $\gdef\vec{\mathbf}$ $\gdef\bv#1{\begin{bmatrix} #1 \end{bmatrix}}$ $\gdef\proj{\operatorname{proj}}$ $\gdef\comp{\operatorname{comp}}$
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1-minute review

Vector Operations

Operation Notation Formula
Magnitude $| \mathbf v |$ $\sqrt{\sum v_i^2}$
Scalar Multiplication $c \mathbf v$ $\left\langle c v_1, \ldots, c v_n \right\rangle$
Vector Addition $\mathbf v + \mathbf w$ $\left\langle v_1 + w_1, \ldots, v_n + w_n \right\rangle$
Dot Product $\mathbf v \cdot \mathbf w$ $\sum v_i w_i$
Cross Product $\mathbf v \times \mathbf w$ $ \langle v_2 w_3 - v_3 w_2, v_3 w_1 - v_1 w_3, v_1 w_2 - v_2 w_1 \rangle $

Odds and ends

  • $\proj_{\vec w} \vec v = \frac{\vec v \cdot \vec w}{\vec w \cdot \vec w}\vec w$
  • projection breaks vectors into (orthogonal) components

The Cross Product

There is a special vector product, or cross product in $\RR^3$ only. \[\vec v \times \vec w = \vec u\] "A vector cross a vector is a vector."

Defining properties
  • bilinearity: $ \vec u \times (c\vec v + d\vec w) = c \vec u \times \vec v + d \vec u \times \vec w$
  • skew-symmetry: $\vec v \times \vec w = - \vec w \times \vec v$
  • $\vec i \times \vec j = \vec k, \qquad \vec j \times \vec k = \vec i, \qquad \vec k \times \vec i = \vec j$

Formula

\[\langle a,b,c \rangle\times \langle d,e,f \rangle = (a\vec i + b\vec j + c \vec k)\times(d\vec i + e\vec j + f \vec k) \]
\[ = (bf - ce) \vec i + (cd - af) \vec j + (ae - bd) \vec k \]
$ = \begin{vmatrix} \vec i & \vec j & \vec k \\a &b &c \\ d & e & f \\ \end{vmatrix} $ $ \begin{matrix} \vec i & \vec j \\a &b \\ d & e \\ \end{matrix} $

Properties of $\vec v \times \vec w$

  • $\vec v \times \vec w$ is perpendicular to both $\vec v $ and $\vec w$.
  • $\vec v \times \vec w$ points in the direction according to the right-hand-rule.
  • $|\vec v \times \vec w| = |\vec v ||\vec w|\sin\theta$ where theta is the (positive) angle between the vectors.

Cross Products and Area

If $\vec v$ and $\vec w$ define adjacent sides of a parallelogram, the area is $|\vec v \times \vec w|$.

Lines

Parametric Form

A line in $\RR^n$ with position $\vec p$ and direction $\vec v$ has parametric form \[ \vec r(t) = \vec p + t\,\vec v \]

Example

Find a parametric form for the line through $(-1, 0, -1)$ and $(1, 1,1)$.

Example

Find position and director vectors for the line with parametric equations: \[ \begin{align*} x &= t - 1 \\ y &= t /2 \\ z &= 3 - 3t \\ \end{align*} \]

Example

Find the intersection of the line through $(0,0)$ and $(2,1)$ and the line with position $\langle 6, -1 \rangle$ and direction $\langle 4, -6 \rangle$.

Solution.

\[ t \bv{2 \\ 1} = \bv{6 \\ -1} + s \bv{4 \\ -6} \]

$t = 2, s = - \frac{1}{2}$ means intersection point is $(4, 2)$.

Follow up. At what angle do these lines meet?

Planes

Forms

We could define a plane via its parametric form with a position vector and 2 direction vectors. \[\vec r(s,t) =\vec p + s \vec v + t \vec w\]

But in $\RR^3$, we can define a plane with a 1 position vector $\vec p$ and one normal vector $\vec n$ to define the plane equation.

\[\vec n \cdot (\vec x - \vec p) = 0\]
Exercise

What is a normal vector to the plane given by \[ x+2y = 16-8z \,?\] Find a point on this plane.

Exercises
  1. Find an equation of the plane through $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$.
    \[x + y + z = 1\]
  2. Find a parametric form for the line of intersection of the planes given by $x+y-z = 2$ and
    $2x - y + 3z = 1$.
    \[\langle 1 + 2t, 1 - 5t, -3t \rangle\]
  1. Find an equation of the plane through the origin and the line \[\vec r(t) = \bv{2 - t \\ t \\ 2t + 1}.\]

Distances

The distance between sets is defined as the minimum of all distances between points in the respective sets. \[ \operatorname{dist}(X,Y) = \operatorname{min}\limits_{\vec x \in X, \vec y \in Y} |\vec x - \vec y| \]

Point-to-Plane

Find the distance from a position $\vec y$ to a plane with normal $\vec n$ and position $\vec p$.

\[ |\proj_{\vec n} (\vec y - \vec p)| \]

Learning Outcomes

You should be able to...
  • Compute and articulate the direction and magnitude of the cross-product in terms of the inputs.
  • Demonstrate that both lines and planes (in dim 3) can be determined with 2 vectors.
  • Contrast the (prescriptive) parametric form with the (descriptive) equation forms of each.
  • Find intersections and generally solve problems of the form "Find the line plane that..."
  • Select the appropriate role of projection in distance problems between points, lines, and planes.