This document summarizes the first lecture of a multivariable calculus course. It introduces key concepts of 3-dimensional space including points, distance, and the rectangular coordinate system. It also covers vectors including their properties, dot products, and cross products. The lecture defines vectors and vector operations in 2D and 3D space. It provides examples of calculating the dot product of vectors and relating it to the angle between vectors. Direction angles and direction cosines of vectors are also introduced.

1. (Multivariable Calculus)
By
ZERTAISHA NASIR
Online Classes Fall 2020
Lecture 1
Department of Mathematics, CUI Wah

2. https://drive.google.com/file/d/1d3Vljw5pFZaXqQxC1SSPpc283KIDrR-
6/view?usp=sharing
Video link of lecture 1

3. Points in 3
dimensions
Distance in
three
dimensions
Rectangular
Coordinate
System
Properties
of Vectors
Dot
Product of
Vectors
Cross
Product of
Vectors
Vectors
Contents
3-Dimensional Space Vectors
Zertaisha Nasir Dept. of Mathematics, CUI Wah

4. Rectangular Coordinate System
Zertaisha Nasir Dept. of Mathematics, CUI Wah

5. Zertaisha Nasir Dept. of Mathematics, CUI Wah

6. Represents a point on YZ-plane
Represents a point on XZ-plane
???
???
???

7. Basic Terms
Practice
Zertaisha Nasir Dept. of Mathematics, CUI Wah

8. Vectors
Zertaisha Nasir Dept. of Mathematics, CUI Wah

9. •-
•-
•-
•-
Zertaisha Nasir Dept. of Mathematics, CUI Wah

10. •-
Zertaisha Nasir Dept. of Mathematics, CUI Wah

11. Practice
Solution

12. Definition:
If and are vectors in 2-space then the dot
product between and is defined as
and is defined as
Similarly, if and are vectors
in 3-Space then the dot product is defined as
)
,
( 2
1 u
u
u 

)
,
( 2
1 v
v
v 


u

v


 v
u
2
2
1
1 v
u
v
u
v
u 




)
,
,
( 3
2
1 u
u
u
u 

)
,
,
( 3
2
1 v
v
v
v 

3
3
2
2
1
1 v
u
v
u
v
u
v
u 





Zertaisha Nasir Dept. of Mathematics, CUI Wah

13.  If u and v are non zero vectors in 2-space or 3-space, and if is the
angle between them, then

cos 
u.v
u v
Zertaisha Nasir Dept. of Mathematics, CUI Wah

15. Direction Angles
In both 2-space and 3-space the angles between a non-zero vector v and the
vector i, j and k are called the direction angles of v, and the consines of
those angles are called the direction cosines of v. Formula for the
direction cosines of a vector can be obtained as, if 1 2 3
ˆ
ˆ ˆ
v v i v j v k
  
Zertaisha Nasir Dept. of Mathematics, CUI Wah

16. Zertaisha Nasir Dept. of Mathematics, CUI Wah

17. Practice

19. https://drive.google.com/file/d/1fMXIKzyxSJrtckFqwadZMduDCAxtCEFO/
view?usp=sharing