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Physics 1D03 - Lecture 3 1
VectorsVectors
• Scalars and Vectors• Vector Components and Arithmetic• Vectors in 3 Dimensions• Unit vectors i, j, k
Serway and Jewett Chapter 3
Physics 1D03 - Lecture 3 2
Physical quantities are classified as scalars, vectors, etc.
Scalar : described by a real number with units
examples: mass, charge, energy . . .
Vector : described by a scalar (its magnitude) and a direction in space
examples: displacement, velocity, force . . .
Vectors have direction, and obey different rules of arithmetic.
Physics 1D03 - Lecture 3 3
Notation
• Scalars : ordinary or italic font (m, q, t . . .)
• Vectors : - Boldface font (v, a, F . . .)
- arrow notation
- underline (v, a, F . . .)
• Pay attention to notation :
“constant v” and “constant v” mean different things!
.) . . F ,a ,v(
Physics 1D03 - Lecture 3 4
Magnitude : a scalar, is the “length” of a vector.
e.g., Speed, v = |v| (a scalar), is the magnitude of velocity v
A A
2
3
A
21
Multiplication:
scalar vector = vector
Later in the course, we will use two other types of multiplication:
the “dot product” , and the “cross product”.
Physics 1D03 - Lecture 3 5
Vector Addition: Vector + Vector = Vector
CBA
e.g.
A B
Triangle Method Parallelogram Method
A
B
A
B
BAC
BAC
Physics 1D03 - Lecture 3 6
Concept Quiz
Two students are moving a refrigerator. One pushes with a force of 200 newtons, the other with a force of 300 newtons. Force is a vector. The total force they (together) exert on the refrigerator is:
a) equal to 500 newtons
b) equal to newtons
c) not enough information to tell
22 300200
Physics 1D03 - Lecture 3 7
Concept Quiz
Two students are moving a refrigerator. One pushes with a force of 200 newtons (in the positive direction), the other with a force of 300 newtons in the opposite direction. What is the net force ?
a)100N
b)-100N
c) 500N
Physics 1D03 - Lecture 3 8
Coordinate Systems
In 2-D : describe a location in a plane
• by polar coordinates :
distance r and angle
• by Cartesian coordinates :
distances x, y, parallel to axes with: x=rcosθ y=rsinθ
x
y
r
( x , y )
0 x
y
Physics 1D03 - Lecture 3 9
Components
• define the axes first
• are scalars
• axes don’t have to be horizontal and vertical
• the vector and its components form a right triangle with the vector on the hypotenuse
) (and , , zyx vvv
x
y
vy
vx
v
Physics 1D03 - Lecture 3 10
3-D Coordinates (location in space)
y
z
x
y
x
z
We use a right-handed coordinate system with three axes:
Physics 1D03 - Lecture 3 11
x
y
z
Is this a right-handed coordinate system?
Does it matter?
Physics 1D03 - Lecture 3 12
Unit Vectors
A unit vector u or is a vector with magnitude 1 :
(a pure number, no units)
Define coordinate unit vectors i, j, k along the x, y, z axis.
1u u
z
y
x
i
j
k
Physics 1D03 - Lecture 3 13
A vector can be written in terms of its components:A
kAjAiAA zyx
i
j
A
Ax i
Ay j
Ay j
Ax i
A
Physics 1D03 - Lecture 3 14
Addition again:
Ax
Ay
A
By
Bx
B
By
Bx
B
Ay
Ax
A
C
Cx
Cy
If A + B = C ,
then:
zzz
yyy
xxx
BAC
BAC
BAC
Three scalar equations from one vector equation!
Tail to Head
Physics 1D03 - Lecture 3 15
CBA
In components (2-D for simplicity) :
jiji )( )( yxyyxx CCBABA
The unit-vector notation leads to a simple rule for the components of a vector sum:
BA
Eg: A=2i+4j B=3i-5j
A+B = 5i-j A - B = -i+9j
Physics 1D03 - Lecture 3 16
Magnitude : the “length” of a vector. Magnitude is a scalar.
In terms of components:
On the diagram,
vx = v cos
vy = v sin x
y
vy
vx
v
e.g., Speed is the magnitude of velocity:
velocity = v ; speed = |v| = v
22|| yx vv v
Physics 1D03 - Lecture 3 17
Summary
• vector quantities must be treated according to the rules of vector arithmetic
• vectors add by the triangle rule or parallelogram rule(geometric method)
• a vector can be represented in terms of its Cartesian components using the “unit vectors” i, j, kthese can be used to add vectors (algebraic method)
• if and only if:
A
BAC
zzzyyyxxx BACBACBAC
