Math 2433 Section 14379 Notes – Week 1
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some of the information you will need for the class.
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Section 14379 LecPop01_1” in the subject line. Be sure to include
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1. What is today?
a. Monday
b. Tuesday
c. Wednesday
d. Thursday
e. Friday
12.1 Cartesian Space Coordinates
Give the equation of a plane that is parallel to the xz-plane that passes
through the point (−1,3, −2) .
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2. The plane x = 3 is parallel to the
a. xy-plane
b. xy-plane
c. xy-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:
r2
( x − a) + ( y − b) + ( z − c) =
2
2
2
Examples:
1) Give the equation of the sphere that has A and B as the endpoints of a
diameter.
A (2, 0, -1) B (2, 1, 3)
2) Find the center and radius of x 2 + y 2 + z 2 + 4 x − 8 y − 2 z + 5 =
0
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:
a+b=b+a
Associative:
(a + b) + c = a + (b + c)
The zero vector 0 = (0,0,0)
(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)
2) Set a = (-5, -2, 6), b = (3, 0, 4), c = (-5, 1, 5). Find: 4a + b - 3c
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3. If you are given two points and wish to find the vector from one to the
other, you would use which operation below?
a. Add
b. Subtract
c. Multiply
d. Divide
e. Vectors cannot be found from points because the notations are too
similar.
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.
3) 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
ua =
ua has direction a
a
4) 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 ) = a1i + a2 j + a3k
5) Calculate the norm of the vector:
7i + 3j - 4k
6) Find α given 3i + j – k and αi – 4j + 4k are parallel
7) Find α so that the norm of αi +( α-1)j + (α+1)k is 2.
8) Find a unit vector in the opposite direction of a = i + 2j - k
9) Find a 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