Math 51 section 3, Tuesday 1/11/2011
Announcements:
1. SUMO tutoring is available starting this week. See http://sumo.stanford.edu/50stutoring.html.
2. Homework is due either in section or by 5pm in an envelope outside my office, 380-N.
I. Dot product
Let v, w be vectors in Rn . Their dot product is
   
w1
v1
 ..   .. 
v · w =  .  ·  .  = v1 w 1 + · · · + vn w n .
vn
wn
It’s a real number!
   
3
2
Example. −1 ·  3  =
1
−5
Selected Properties:
p
• Let kvk = v12 + · · · + vn2 be the length of v. Then v · v = kvk2 .
• Let θ be the angle between v, w 6= ~0. Then v · w = kvkkwk cos(θ).
• For v, w 6= ~0, we have v · w = 0 if and only if θ = 90◦ .
2
1
Example.
·
=
−1
2
• By taking absolute values above, we get that
|v · w| = kvkkwk | cos(θ)| ≤ kvkkwk.
This is called the Cauchy-Schwarz Inequality.
Problem 1 (LA 4.20(b)). Show that (v + w) · x = v · x + w · x for any vectors v, w, x ∈ Rn .
1
II. Cross product
Let v, w be vectors in R3 . Their cross product is


v2 w 3 − v3 w 2
v × w = v3 w1 − v1 w3 
v1 w 2 − v2 w 1
It’s a vector in R3 !
   
3
2
Example. −1 ×  3  =
1
−5
Selected Properties:
• v and v × w are orthogonal. So are w and v × w. That is,
v · (v × w) = ~0 = w · (v × w).
This tells you the direction of v × w, up to a sign.
• Let θ be the angle between v, w 6= ~0. The magnitude of v × w is given by
kv × wk = kvkkwk sin(θ),
which is the area of the parallelogram formed by v and w.
 
 
 
−1
0
2
Problem 2. Let u = 0, v =  2 , and w = 2.
4
1
1
(a) Using the cross product, find a normal vector to the plane containing these three points.
(b) Using (a), find a non-parametric equation for this plane involving x, y, and z.
(c) Find a parametric representation of the plane containing these three points.
III. Reduced row echelon form, pivot variables, free variables
Problem 3. Solve the following system of linear equations by first putting its augmented matrix into reduced
row echelon form. Which are the free and pivot variables?
2x + 4y + 2z = 2
−2x − 3y + z = 2
3x + 5y = −1
2