1. Vectors and Scalars

3. Topic 1: Measurement and uncertainties 1.3 – Vectors and scalars
Essential idea: Some quantities have direction and magnitude, others have magnitude only, and this understanding is the key to correct manipulation of quantities. This sub-topic will have broad applications across multiple fields within physics and other sciences.
Nature of science: Models: First mentioned explicitly in a scientific paper in 1846, scalars and vectors reflected the work of scientists and mathematicians across the globe for over years on representing measurements in three dimensional space.
© 2006 By Timothy K. Lund

4. Topic 1: Measurement and uncertainties 1.3 – Vectors and scalars
Understandings:
• Vector and scalar quantities
• Combination and resolution of vectors
Applications and skills:
• Solving vector problems graphically and algebraically
© 2006 By Timothy K. Lund 4

5. Topic 1: Measurement and uncertainties 1.3 – Vectors and scalars
Guidance:
• Resolution of vectors will be limited to two perpendicular directions
• Problems will be limited to addition and subtraction of vectors and the multiplication and division of vectors by scalars
© 2006 By Timothy K. Lund 5

6. Topic 1: Measurement and uncertainties 1.3 – Vectors and scalars
International-mindedness:
• Vector notation forms the basis of mapping across the globe
Theory of knowledge:
• What is the nature of certainty and proof in mathematics?
Utilization:
• Navigation and surveying (see Geography SL/HL syllabus: Geographic skills)
• Force and field strength (see Physics sub-topics 2.2, 5.1, 6.1 and 10.1)
• Vectors (see Mathematics HL sub-topic 4.1; Mathematics SL sub-topic 4.1)
Aims:
• Aim 2 and 3: this is a fundamental aspect of scientific language that allows for spatial representation and manipulation of abstract concepts
© 2006 By Timothy K. Lund 6

7. A SCALAR is ANY quantity in physics that has MAGNITUDE.
Magnitude – A numerical value with units.
A VECTOR is ANY quantity in physics that has BOTH MAGNITUDE and DIRECTION.
Scalar
Magnitude
Speed
20 m/s
Distance
10 m
Age
15 years
Energy
1000 J
Power
25 W
Vector
Magnitude & Direction
Displacement
35 m, 300
Velocity
20 m/s, N
Acceleration
10 m/s/s, E
Force
5 N, West

8. The difference in scalars and vectors is in algebra.
▪ Scalars obey rules of ordinary algebra.
▪ Vectors are quantities that need direction for full description.
As such, the addition of two or more vectors must take into account
their directions. Vectors obey rules of vector algebra.
When two forces are acting on an object, for example 5N and 6N, the net force, will depend on directions of 5N force and 6N force. Adding these two vectors can result in anything between 1N and 11N net force, depending on their directions.
The rules for adding vectors are different than the rules for adding two scalars, for example 2kg potato + 2kg potatos = 4 kg potatoes. Mass doesn’t have direction.

9. Topic 1: Measurement and uncertainties 1.3 – Vectors and scalars
PRACTICE:
Unfortunately weight is sometimes considered as vector (gravitational force) and sometimes as scalar (magnitude of gravitational force)
© 2006 By Timothy K. Lund 9

10. The length of the vector, drawn to scale, indicates the magnitude of the vector quantity.
the direction of a vector is the counterclockwise angle of rotation which that vector makes with due East or x-axis.

11. Two vectors are equal if they have the same magnitude and the same direction.
This is the same vector. It doesn’t matter where it is. You can move/translate it around, . It is determined ONLY by magnitude and direction, NOT by starting point.

12. Multiplying vector by a scalar
Multiplying a vector by a scalar will ONLY CHANGE its magnitude. A
½ A
Multiplying vector by 2 increases its magnitude by a factor 2, but does not change its direction.
One exception:
Multiplying a vector by “-1” does not change the magnitude, but it does
reverse it's direction
Opposite vectors A
– 3A
– A
- A

13. Adding vectors:
 Consider two vectors drawn to scale: vector 𝐴 and vector 𝐵 .
 In print, vectors are designated in bold non-italicized print: A, B.
When doing problems, label vectors as: 𝐴 and 𝐵
We take the second-named vector 𝐵 , and translate it towards the first-named vector 𝐴 , so that 𝐵 ’s TAIL connects to 𝐴 ’s TIP.
The result of the sum, which we are calling the vector 𝑺 (for sum), is gotten by drawing an arrow from the START of 𝐴 to the FINISH of 𝐵 . 𝑺 𝐵 𝐵 𝐴

14. The order in which two or more vectors are added does not effect result.
vectors can be moved around as long as their length (magnitude) and direction are not changed.
Vectors that have the same magnitude and the same direction are the same.
RESULTANT

15. Topic 1: Measurement and uncertainties 1.3 – Vectors and scalars
Combination and resolution of vectors
PRACTICE:
SOLUTION:
Resultant is another word for sum.
Draw the 7 N vector, then from its tip, draw a circle of radius 5 N:
Various choices for the 5 N vector are illustrated, together with their vector sum:
© 2006 By Timothy K. Lund
The shortest possible vector is 2 N. 15

16. the opposite of the vector B
 SUBTRACTION is adding opposite vector.
- B B
the vector B
A + - B A
- B
the opposite of the vector B
Thus,
A - B = A +
- B

17. Topic 1: Measurement and uncertainties 1.3 – Vectors and scalars
Combination and resolution of vectors
SOLUTION:
Sketch the sum.
© 2006 By Timothy K. Lund y
c = x + y x 17

18. Topic 1: Measurement and uncertainties 1.3 – Vectors and scalars
SOLUTION:
Sketch in the difference.
© 2006 By Timothy K. Lund
𝒁 = 𝑿 − 𝒀 𝑿
− 𝒀 18

19. Components of Vectors q
To resolve a vector means to break it down into its x- and y-components.
Ax = A cos 
Ay = A sin 
y – component of the vector
Vertical component A Ay q
if the vector is in the first quandrant;
if not, find  from the picture. Ax
Horizontal component
x – component of the vector
Data booklet reference:
• AH = A cos 
• AV = A sin 

20. q v = 34 m/s @ 48° . Find vx and vy vx = 34 m/s cos 48° = 23 m/s wind
Examle: A plane moves with velocity of 34 48°.
Calculate the plane's horizontal and vertical velocity components.
We could have asked: the plane moves with velocity of 34 48°.
It is heading north, but the wind is blowing east.
Find the speed of both, plane and wind.
v = 34 48° . Find vx and vy vy q vx
vx = 34 m/s cos 48° = 23 m/s wind
vy = 34 m/s sin 48° = 25 m/s plane

21. If you know x- and y- components of a vector you can find the magnitude and direction of that vector:
Let:
Fx = 4 N
Fy = 3 N .
Find magnitude (always positive) and direction. Fy q
 = arc tan (¾) = 370 Fx

22. Vector addition – analyticallyBy Cy Bx Ay Ax Cx

23. example:
= 68 24°
= 32 65°
Fx = F1x + F2x = 68 cos cos650 = 75.6 N
Fy = F1y + F2y = 68 sin sin650 = 56.7 N
 = arc tan (56.7/75.6) = 36.90

24. Topic 1: Measurement and uncertainties 1.3 – Vectors and scalars
Combination and resolution of vectors
As a more entertaining example of the same technique, let us embark on a treasure hunt.
Arrgh, matey. First, pace off the first vector A.
Then, pace off the second vector B.
© 2006 By Timothy K. Lund
And ye'll be findin' a treasure, aye!

25. Topic 1: Measurement and uncertainties 1.3 – Vectors and scalars
HOW to ADD any vector
We can think of the sum A + B = S as the directions on a pirate map.
We start by pacing off the vector A, and then we end by pacing off the vector B.
S represents the shortest path to the treasure.
© 2006 By Timothy K. Lund B
end A S A + B = S
start