1. Vector Review Scalars are magnitudes without direction.
(i.e. temperature…What’s the direction of temperature? This is a meaningless question.)
Vectors have both a magnitude & direction.
(i.e. velocity is a speed in a particular direction, where the speed is the scalar quantity of the velocity)

2. The Point of Vectors
The concept of direction is rooted in the spatial dimensionality of the system. For instance, in a 2-D space, position can be described by a set of two variables, or degrees of freedom: y s x
Chp 3

3. A unit vector has Direction A variable length A set length 1 and 2
All 3

4. The Point of Vectors
The concept of direction is rooted in the spatial dimensionality of the system. For instance, in a 2-D space, position can be described by a set of two variables, or degrees of freedom: y
Begin by defining the unit vectors, which determine the directions of the basis vectors of the space. The unit vectors are:
orthogonal (a.k.a. perpendicular)
of unit length (meaning they have a magnitude of 1) s x

5. The Point of Vectors
The concept of direction is rooted in the spatial dimensionality of the system. For instance, in a 2-D space, position can be described by a set of two variables, or degrees of freedom: y s x sx

6. The Point of Vectors
The concept of direction is rooted in the spatial dimensionality of the system. For instance, in a 2-D space, position can be described by a set of two variables, or degrees of freedom: y y
S = <sx, sy> sy s s x x sx

7. The Point of Vectors
The concept of direction is rooted in the spatial dimensionality of the system. For instance, in a 2-D space, position can be described by a set of two variables, or degrees of freedom: y y
S = <sx, sy> sy s s x x sx r

8. The Point of Vectors
The concept of direction is rooted in the spatial dimensionality of the system. For instance, in a 2-D space, position can be described by a set of two variables, or degrees of freedom: y y
S = <sx, sy>
S = <r, f> sy s s r f x x sx

9. The Point of Vectors Another notation: S = <sx, sy>
S = <r, f> y y sy s s r f x x sx

10. The Point of Vectors S = <r, f> = <sx, sy> y Look to Trig:
r2 = sx2 + sy2
sx = r cosf
sy = r sinf
tanf = sy/sx
S = <r, f> = <sx, sy> s r sy f x sx
But since /s/ = /r/,
s2 = sx2 + sy2
LBCC Physical Sciences Dept. Fall 2006
PH 211: General Physics w/ Calculus

11. The Point of Vectors sx y Vector Addition: x Start w/ sx, sx
(a geometric approach) sx x
Start w/ sx, sx
LBCC Physical Sciences Dept. Fall 2006
PH 211: General Physics w/ Calculus

12. The Point of Vectors sx + sy y Vector Addition: sy x Start w/ sx,
(a geometric approach)
sx + sy sy x
Start w/ sx,
Add the tail of sy to the head of sx, sx
LBCC Physical Sciences Dept. Fall 2006
PH 211: General Physics w/ Calculus

13. The Point of Vectors
So far, we have been dealing with a particular vector in terms of its vector components, but these methods are also applicable for generic vectors: A A
LBCC Physical Sciences Dept. Fall 2006
PH 211: General Physics w/ Calculus

14. The Point of Vectors
So far, we have been dealing with a particular vector in terms of its vector components, but these methods are also applicable for generic vectors:
A + B B A
LBCC Physical Sciences Dept. Fall 2006
PH 211: General Physics w/ Calculus

15. The Point of Vectors
So far, we have been dealing with a particular vector in terms of its vector components, but these methods are also applicable for generic vectors: C
A + B = C B A
LBCC Physical Sciences Dept. Fall 2006
PH 211: General Physics w/ Calculus

16. The Point of Vectors Your Turn: Which figure shows A1 + A2 + A3?
Your choices are the following vectors, all of equal magnitude:
1) horizontal, pointing left 2) 3)
4) vertical, pointing down 5)
6) horizontal, pointing right A1 A3 A2

17. The Point of Vectors
What about subtraction? C
A + B = C B A
Chp. 3

18. The Point of Vectors Additional Comments on Vector Algebra:
There is no absolute coordinate system. This means you are free to choose whichever coordinate system at whichever orientation best suits your needs. Choose wisely in order to simplify your problems--the best choice of coordinate system typically will reduce the number of vectors you will need to find components for and therefore will simplify your problem.