1. Serway and Jewett Chapter 3
Vectors
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. Vectors have direction, and obey different rules of arithmetic.
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!
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
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
Triangle Method
Parallelogram Method
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:
equal to 500 newtons
equal to newtons
not enough information to tell
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) 100 N
B) -100 N
C) 500 N
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 y  x x
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 x y vy vx
Physics 1D03 - Lecture 3

10. Magnitude : the “length” of a vector. Magnitude is a scalar.
e.g., Speed is the magnitude of velocity:
velocity = v ; speed = |v| = v
In terms of components:
On the diagram,
vx = v cos 
vy = v sin  x y vy vx 
Physics 1D03 - Lecture 3

11. 3-D Coordinates (location in space)
We use a right-handed coordinate system with three axes: y z x y x z
Physics 1D03 - Lecture 3

12. Is this a right-handed coordinate system?x y z
Is this a right-handed coordinate system?
Does it matter?
Physics 1D03 - Lecture 3

13. 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. z y x i j k
Physics 1D03 - Lecture 3

14. A vector can be written in terms of its components:
Ay j
Ay j j
Ax i i
Ax i
Physics 1D03 - Lecture 3

15. Addition again: Ay By Ax If A + B = C , Bx then: Tail to Head Cy
Three scalar equations from one vector equation! Ay Ax Cx
Physics 1D03 - Lecture 3

16. In components (2-D for simplicity) :
The unit-vector notation leads to a simple rule for the components of a vector sum:
In components (2-D for simplicity) :
Eg: A=2i+4j B=3i-5j
A+B = 5i-j A - B = -i+9j
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, k these can be used to add vectors (algebraic method)
if and only if:
Physics 1D03 - Lecture 3