Transcript Document

Vectors
Scalars: a physical quantity described by a single number
Vector: a physical quantity which has a magnitude (size) and
direction.
•Examples: velocity, acceleration, force, displacement.
•A vector quantity is indicated by bold face and/or an arrow.


a or a or a or a etc
•The magnitude of a vector is the “length” or size (in
appropriate units).


a  a (the magnitude of a )
The magnitude of a vector is always positive.
•The negative of a vector is a vector of the same magnitude
put opposite direction (i.e. antiparallel)
Phys211C1V p1
Combining scalars and vectors
scalars and vectors cannot be added or subtracted.
the product of a vector by a scalar is a vector
x=ca
x = |c| a (note combination of units)
if c is positive, x is parallel to a
if c is negative, x is antiparallel to a
Phys211C1V p2
Vector addition
most easily visualized in terms of displacements
Let X = A + B + C
graphical addition: place A and B tip to tail;
X is drawn from the tail of the first to the tip
of the last
B
A
X
A+B=B+A
X
A
B
Phys211C1V p3
Vector Addition: Graphical Method of R = A + B
•Shift B parallel to itself until its tail is at the head of A,
retaining its original length and direction.
•Draw R (the resultant) from the tail of A to the head of B.
B
B
A
+
= A
=
R
the order of addition of several vectors does not matter
C
C
D
B
B
C
A
B
A
D
A
D
Phys211C1V p4
Vector Subtraction: the negative of a vector points in the opposite
direction, but retains its size (magnitude)
-B
• A- B = A +( -B)
R
A
-
B
= A
+
-B
=
A
Phys211C1V p5
Resolving a Vector (2-d)
replacing a vector with two or more (mutually perpendicular)
vectors => components
directions of components determined by coordinates or geometry.
A = Ax + Ay
Ax = x-component
Ay = y-component
A  Ax + Ay
2
tan q 
Ay
Ax
A
Ay
q
2
Ax  A cosq
Ax
Ay  A sin q
Be careful in 3rd , 4th quadrants when using inverse
trig functions to find q.
Component directions do not have to be horizontalvertical!
A
Ay
q
Ax
Phys211C1V p6
Vector Addition by components
R=A+B+C
Resolve vectors into components(Ax, Ay etc. )
Add like components
Ax + Bx + Cx = Rx
Ay + By + Cy = Ry
The magnitude and direction of the resultant R can be determined
from its components.
in general R  A + B + C
Example 1-7: Add the three displacements:
72.4 m, 32.0° east of north
57.3 m, 36.0° south of west
72.4 m, straight south
Phys211C1V p7
Unit Vectors
a unit vector is a vector with magnitude equal to 1 (unit-less and
hence dimensionless)
in the Cartesian coordinates:
iˆ unit vecto r in + x direction
ˆj unit vecto r in + y direction
kˆ unit vecto r in + z direction
Right Hand Rule for relative directions: thumb, pointer, middle
for i, j, k.
Express any vector in terms of its components:

A  Ax iˆ + Ay ˆj + Az kˆ
Phys211C1V p8
Products of vectors (how to multiply a vector by a vector)
Scalar Product (aka the Dot Product)
 
 
 
A  B  AB cos   A B cos   B  A
0    180
is the angle between the vectors
A.B = Ax Bx +Ay By +Az Bz = B.A
= B cos  A is the portion of B along A times the magnitude of A
= A cos  B is the portion of A along B times the magnitude of B
B
A

B cos
note: the dot product between perpendicular vectors is zero.
ˆi  ˆi  1 ˆj  ˆj  1 kˆ  kˆ  1
ˆi  ˆj  0 ˆj  kˆ  0 kˆ  ˆi  0
Phys211C1V p9
Example: Determine the scalar product between
A = (4.00m, 53.0°) and B = (5.00m, 130.0°)
Phys211C1V p10
Products of vectors (how to multiply a vector by a vector)
Vector Product (aka the Cross Product) 3-D always!
  
 
C  A  B  -B  A
C  AB sin 
is the angle between the vectors
0    180
Right hand rule: AB = C
A – thumb
B – pointer
C – middle
Cartesian Unit vectors
ˆi  ˆi  0 ˆj  ˆj  0 kˆ  kˆ  0
ˆi  ˆj  kˆ ˆj  kˆ  ˆi kˆ  ˆi  ˆj
C = AB sin 
= B sin A is the portion of B perpendicular A times the magnitude of A
= A sin B is the portion of A perpendicular B times the magnitude of B
Phys211C1V p11
  
 
C  A  B  -B  A
C  AB sin 
C = AB sin 
= B sin  A is the part of B perpendicular A times A
= A sin  B is the part of A perpendicular B times B
B sin
B
A

Write vectors in terms of components to calculate cross product
 
A  B  ( Ax iˆ + Ay ˆj + Az kˆ )  ( Bx iˆ + B y ˆj + Bz kˆ )
 Ax iˆ  Bx iˆ + Ax iˆ  B y ˆj + Ax iˆ  Bz kˆ
+ Ay ˆj  Bx iˆ + Ay ˆj  B y ˆj + Ay ˆj  Bz kˆ
+ Az kˆ  Bx iˆ + Az kˆ  B y ˆj + Az kˆ  Bz kˆ
 ( Ay Bz - Az B y )iˆ + ( Az Bx - Ax Bz ) ˆj + ( Ax B y - Ay Bx )kˆ
Phys211C1V p12
Example: A is along the x-axis with a magnitude of 6.00 units, B
is in the x-y plane, 30° from the x-axis with a magnitude of 4.00
units. Calculate the cross product of the two vectors.
Phys211C1V p13