Download Access code deadline 6/14

Access code deadline 6/14
Math 2433 Notes – Week 1 Session 1
Welcome! We will start at 6:00. We will start off by going over some of the information you will
need for the class.
• Please log on to all sessions with firstname lastname
• CASA – www.casa.uh.edu – If you do not have access yet, email the CASA tech support
(name, id and class included in email). Note: if you registered late for the class, it takes a few
days for you to be listed on CASA rolls. Also, for this week only, if you do not have access
then email me your popper answers after class with Math 2433 Popper 1 in title. Be sure to
include your name and id in the email.
• If you miss all lecture session options then you will need to do the poppers that are in these
notes before Saturday night at 11:59pm.
• No homework will be accepted through email!!!!!!
• No late homework accepted!!!! (Do not wait until last hour to upload homework and
alternates – be sure to submit a minimum of 2 hours before deadline)
• I do NOT drop any quiz grades.
• Quiz 1 – 5 are open and will close June 13th.
• Watch posted videos before coming to class.
12.1 Cartesian Space Coordinates
Ex: Give the equation of a plane that is parallel to the xz-plane that passes through the point (2, -1, 4).
Popper 1
1. The plane z = 2 is parallel to the
a. xy-plane
b. yz-plane
c. xz-plane
Distance Formula: d ( P1 , P2 ) = ( x2 − x1 )2 + ( y2 − y1 )2 + ( z2 − z1 )2
⎛ x + x y + y2 z1 + z2 ⎞
Midpoint Formula: ⎜ 1 2 , 1
,
⎟
2
2 ⎠
⎝ 2
Equation of a Sphere:
( x − a ) + ( y − b) + ( z − c )
2
2
2
= r2
Examples:
1) Give the equation of the sphere that has A and B as the endpoints of a diameter.
A (2, 1, 0)
B (1, 1, -3)
2) Find the center and radius of x 2 + y 2 + z 2 + 4 x − 8 y − 2 z + 5 = 0
Popper 1
2. Which of the following describes a sphere with radius of 2 and center at the origin together
with its interior?
a. Ω = {( x, y, z ) : x 2 + y 2 + z 2 ≥ 4}
b. Ω = {( x, y, z ) : x 2 + y 2 + z 2 ≥ 2}
c. Ω = {( x, y, z ) : x 2 + y 2 + z 2 ≤ 4}
d. Ω = {( x, y, z ) : x 2 + y 2 + z 2 ≤ 2}
e. None of the above
12.3 Vectors
A vector is an ordered triple (in space) where addition and multiplication by scalars holds. Vectors
have a direction and a length (magnitude or norm).
Properties of vectors:
Commutative:
Associative:
The zero vector 0 = (0,0,0)
a+b=b+a
(a + b) + c = a + (b + c)
(note: a ⋅ 0 = 0 )
Vectors can be multiplied by a scalar: if a = ( a1 , a2 , a3 ) , then 2a = ( 2a1 ,2a2 ,2a3 )
The norm of a vector a = ( a1 , a2 , a3 ) is a = a12 + a22 + a32
Examples:

1) Find the vector PQ and determine its norm given points P and Q.
P(5,3,2), Q(-3,1,5)
Popper 1
3. Calculate the norm of the vector: (2,1,−2)
a. ½
b. 3
c. 2
d.
5
e. None of the above
2) Set a = (-5, -2, 6), b = (3, 0, 4), c = (-5, 1, 5). Find: 4a + b - 3c
3) Simplify the linear combination:
4(2j - 3k) + 2(2i + 3j - 4k)
Two vectors are parallel if a = α b for some real number α .
If α >0, then a and b have the same direction.
If α <0, then a and b have opposite directions.
4) Are any of the following vectors parallel?
a = (1, -1, 2) b = (2, -1, 2) c = (3, -3, 6) d = (-2, 2, -4)
Unit Vectors are vectors of norm 1.
a
u a has direction a
ua =
a
5) Find the unit vector for a = (3,4,-2)
There are 3 special unit vectors:
i = (1,0,0)
j = (0,1,0)
k = (0,0,1)
All vectors can be represented by a linear combination of these:
( a1, a2 , a3 ) = a1 i + a2 j + a3 k
Why?
6) Calculate the norm of the vector:
7i + 3j - 4k
7) Find α given 3i + j – k and αi – 4j + 4k are parallel
8) Find α so that the norm of αi +( α-1)j + (α+1)k is 2.
9) Find the vector of norm 2 in the opposite direction of a = i + 2j - k
10) Let a = (7, 5, 2), b = (6, 4, 1), c = (7, 5, 7), and d = (4, 4, 6).
Find scalars A, B, C such that d = Aa + Bb + Cc
12.4 The Dot Product
Given a = (a1, a2, a3) and b = (b1, b2, b3), a i b = (a1)(b1) + (a2)(b2) + (a3)(b3)
Note that this gives an answer that is NOT a vector. The dot product gives an answer that is a scalar.
Examples:
1) a = (-1, -3, 5), b = (2, 3, -4)
2) a = 6i + 5j + 4k, b = i + 3j
ai b =
aib =
The angle between two vectors is found with this formula: cosθ =
3) Given, a = 4i + 4j, b = i - 2j + 3k, c = 2i + 2k
Find the angle between a and b
Find the angle between a and c
aib
a b
Popper 1
4. The angle between a = (2, −1,1) and b = (−1, 2,1) is:
a.
b.
c.
d.
e.
π
3
5π
6
3π
4
2π
3
None of the above
Projection of a on b projba = (a i ub )u b where a i ub = compba
4) Given a = 4i + 3j, b = i - 3j + 2k, c = 2i + 4k
Find the projection of a in the b direction.
Find the component of a in the c direction.
The angles α , β , γ that a vector makes with unit vectors i, j, and k are called direction angles of a.
A unit vector with these direction angles is: cos α i + cos β j + cos γ k
5) Find a unit vector with direction angles: α = π⁄6, β = π⁄3, γ = 3π⁄2
6) Find the direction angles of a = i -
3k
7) Find all numbers x such that
2i + 4j + 2xk ⊥ 6i + 3j - 4xk
12.5 Cross Product
If vectors a and b are NOT parallel, they form two sides of a parallelogram:
a
b
a × b is the vector perpendicular to this plane.
So, if a || b then a × b = 0
The area of this parallelogram is A= || a × b ||
for a = ( a1 , a2 , a3 ) and b = (b1 , b2 , b3 ) , a × b =
Example
1) Calculate (3i + j + k) × (i – 2j)
2) Find two unit vectors perpendicular to a = (1,2,-1) and b = (1,0,2)
3) Find the area of the parallelogram with vertices:
A(2, 1, 4), B(1, 4, 3), C(1, 0, 2), D(2, -3, 3)
Volume of a Parallelepiped: V= |(a × b) i c |
4) Find the volume of the parallelepiped with the given edges.
i - 3j + k, 3j - k, i + j - 3k
Popper 1
5. Find all numbers x for which 2i + 5 j + 2 xk ⊥ 6i + 4 j − xk
a. x = 0, 4
b. x = 4
c. x = ±4
d. x = 1
e. None of the above