Vectors
Scalar: Ch3: just a number mass (m) time (t) pressure (P) temperature (T) energy (E) ... Vector: magnitude + direction displacement r
velocity v acceleration
a
force F momentum p ...
Adding Vectors
c = a +b
b
a
c a
c
b
To add vectors, you can slide them as you wish as long as you preserve their direction
Adding Vectors: examples
c = a +b
a a b
a=3 b=4 120
b
a=2 b=2
NOTE: a b
Need to add vectors with different magnitudes, directions!
Vector Components
y
ax = a cos a y = a sin
ay
a
ax
x
2 2 a 2 = ax + a y
a = ax + a y
Components depend on your choice of the coordinate system
You need to know:
cos 0 = 1 sin 0 = 0 cos 30 = sin 30 = 3 2
cos x 0
360 = 2 radians
sin x, cos x look like this:
1 2 1 cos 60 = 2
sin 2 x x (rad)
3 sin 60 = 2 cos 90 = 0 sin 90 = 1
sin (x y) = sin x cos y cos x sin y cos(x y)= cos x cos y sin x sin y
... etc. See Appendix E in HRW
Unit Vectors
y
ay
a
^ i
^ | i | =1 j | ^ | =1
ax
x
^ j
^ a = ax i + a y ^ j
Example:
^ v = (2, 4, -1) = 2i + 4 ^ - k j ^
Adding Vector using Components
y
c = a +b
b
a
cx = ax + bx
c y = a y + by sin = cy c
x
c = c +c
2 x
2 y
Example
y
c = a +b
a
a = 2, b = 4, = 30
x
b
^ j A : c = 5.73i + ^
Example
Multiplying Vectors
Scalar product (dot-product):
a ib = ab cos = ax bx + a y by + az bz
Vector product (cross-product):
b
c = a b c = ab sin
a
c (a , b )-plane, direction: use right hand rule
Examples
1
^ a =i ^ b = -i + 3 ^ j a b = -1 ^ a b = 3 k
b
3
1
a