The document discusses 3D coordinate systems and vectors. It defines: 1) The distance formula in 3D space between two points (x1,y1,z1) and (x2,y2,z2) as the square root of (x2-x1)2 + (y2-y1)2 + (z2-z1)2. 2) Vectors as quantities that have both magnitude and direction. It shows examples of vector addition and subtraction using the triangle and parallelogram laws. 3) Key properties of vectors including that vector addition is commutative and distributive over vector addition, scalar multiplication distributes over vector addition, and vectors

1. 12.1 3D coordinate
systems
z
(a,b,c)
c
o
b x
a
y

2. What is the meaning of the following
equation in R ?
=
=
=
+ + =

3. Distance formula in 3D space:
The distance between ( , , ) and ( , , ) is
| |= ( ) +( ) +( )
An equation of a sphere with center ( , , ) and
radius is
( ) +( ) +( ) =

4. 12.2 Vectors
A vector is a quantity that has both
magnitude and direction.
B
=
A
The zero vector has no specific direction.

5. D
B
v
u
C
A

6. D
B
v
u =
C
=
A

7. D
B
v
u =
C
=
A
u

8. D
B
v
u =
C
=
A
2u
u

9. D
B
v
u =
C
=
A
2u
u
u/2

10. D
B
v
u =
C
=
A
2u
u -u
u/2

11. Vector Addition
v
u

12. Vector Addition and Subtraction
v
u
u v

13. Vector Addition and Subtraction
v
u
u v The Triangle Law
u+v

14. Vector Addition and Subtraction
v
u
u v The Triangle Law
u+v
u
v

15. Vector Addition and Subtraction
v
u
u v The Triangle Law
u+v
u
v

16. Vector Addition and Subtraction
v
u
u v The Triangle Law
u+v
u
u+v The Parallelogram Law
v

17. Vector Subtraction
v
u

18. Vector Subtraction
v = +( )
u

19. Vector Subtraction
v = +( )
u
-v
u

20. Vector Subtraction
v = +( )
u
-v
u-v
u

21. Properties of vectors:
, , are vectors, , are scalars.
+ = +
+( + )=( + )+
+ =
+( )=
( + )= +
( + ) = +
( ) = ( )
=