MACT 1122: Calculus II
Instructor: Wafik Lotfallah
Course Material
 Calculus, Early Transcendentals, by James
Stewart, Metric Version, 8th Edition, 2016
 Supplementary Material on the Blackboard
Prerequisites:
 Formally: MACT 1121
 Informally: Chapters 1-5 from the text.
How is my grade determined?
• Classwork: 10%. Participate in class and solve the
class worksheets.
• Online “Webassign” Homework Assignments: 10%.
(Late assignments will NOT be accepted.)
• Two Wolfram Alpha Assignments: 1% each.
• Two Midterms: 25% each. Both are counted.
• Comprehensive Final: 30%.
How do I ace this course?
 Read the section of the textbook ahead of time.
 Participate in class and solve the class worksheets.
 Take notes in the lecture (just copy the board).
 Solve all homework problems.
 Ask the lecturer and/or the TA during office hours.
 Review the material before each exam.
 Take good sleeps at night.
Vectors & the 3-Dimensional Space
Sections 12.1, 12.2
Lecture 1 Objectives
 Find the distance between two points in the space.
 Find an equation of a given sphere.
 From the sphere equation, find its center and radius.
 Find the sum, difference, scalar multiple, magnitude
of vectors, and the vector joining two given points.
 Find the direction (a unit vector) of a given vector.
 Find a vector given its magnitude and direction.
 Solve velocity word problems using vectors.
The 3D
Space
The 3D
Space
The 3D
Space
Points in
Space
Example
In the 3D space, describe the set of points (x, y, z) for
which:
z=0
Distance
Between
Points
Equations
of Spheres
Examples
1) Find an equation of the sphere that passes through
the point (2, 0, 5) and has center (3, 1, 0).
2) Show that the equation x2 + y2 + z2 + 4x  6y = 3
represents a sphere, and find its center and radius.
How to determine collinearity in three dimensions?
Find if three points in 3-dimensional space are collinear i.e. lie on one straight
line.
Method 1:
A,B,C are collinear if and only if the largest of the lengths of AB,AC,BC is equal
to the sum of the other two.
Method 2:
Point A,B, and C determine two vectors 𝐴𝐵 and 𝐴𝐶 . Suppose the latter is not zero
vector, see if there is a constant 𝜆 that allows 𝐴𝐵= 𝜆 𝐴𝐶 or 𝐴𝐵 × 𝐴𝐶 = 0
Vectors (Physicist’s Definition)
A vector 𝐴𝐵 is a directed line segment.
Equality of Vectors
 Vectors of the same length and direction are equal.
 Question: What is the direction of a vector?
Physicist’s Addition of Vectors,..
Triangle Law
the Parallelogram Law
.. Scalar Multiplication, ..
and Subtraction
We define the difference of two vectors as:
(a) the Parallelogram Law
(b) Triangle Law
Example
Let 𝑂 be the origin. Show that the vector that starts
at the point 𝐴 and ends at the point 𝐵 can be given
by:
𝐴𝐵 = 𝑂𝐵 − 𝑂𝐴
(a)
(b)
(c)
(d)
(e)
u+v
u+w
v+w
u−v
v+u+w
The Mathematician’s Definition
 Bringing the vector 𝐴𝐵 in the standard position 𝑂𝑃,
we define the vector 𝐴𝐵 = 𝑂𝑃 to be the point P.
 We write 𝐴𝐵 = a1, a2, a3.
Mathematician’s Addition,..
Definition: a + b = a1, a2, a3 + b1, b2, b3
= a1 + b1, a2 + b2, a3 + b3
and Scalar Multiplication.
Definition:
ca = ca1, a2, a3
= ca1, ca2, ca3
Basic Properties
Proofs using:
 Algebra
 Geometry
Application: Finding 𝐴𝐵
Example: Are the points
A(1, 2, 3), B(2, 3, 5), and C(3, 4, 6)
collinear (i.e. lie on a straight line)?
An Alternative Notation
where
The Length of a Vector
 Definition: The magnitude (length) of a vector a
is defined by:
𝑎1 , 𝑎2 , 𝑎3
=
𝑎12 + 𝑎22 + 𝑎32
 Example: Show that for any vector a and scalar c,
1) |ca| = |c||a|
2) |a| = 0 if and only if a = 0 = 0, 0, 0.
 Question: When does the following hold?
|a + b| = |a| + |b|
A-b= (2,-3,4)-(-2,1,1) = (2+2, -3-1, 4-1) = (4,-4,3).
3B = 3(-2,1,1) = (-6, 3,3).
2a + 5b = 2(2, -3, 4) + 5(-2,1,1) = (4,-6,8) + (-10, 5,5) = (-6,-1,13).
Unit Vectors
 Definition: A vector u of unit length, i.e. |u| = 1, is
called a unit vector.
 Show that if a  0, then the unit vector u in the
direction of a is given by:
1
𝐮=
a
|a|
 The unit vector u above is called the direction of a.
 Thus, the equation a = a 𝐮 expesses the vector a as
the product of its magnitude and direction.
Thank you for listening.
Wafik
Who are these?
They are the 3 Bs:
Johann Sebastian Bach
(1685-1750 )
Ludwig van Beethoven
(1770-1827 )
Johannes Brahms
(1833-1897)