1. Scalars & Vectors – Learning Outcomes
Differentiate between scalars and vectors.
Recognise quantities as either scalars or vectors.
Add vectors.
Use coordinates to represent and work with vectors.
Decompose vectors into components.
Calculate the direction of vectors.
Solve problems about vectors.

2. Differentiate between scalars and vectors
Scalars are quantities with magnitude only, e.g.
distance,
time,
speed,
temperature,
mass,
For scalars, only magnitude matters.

3. Differentiate between scalars and vectors
Vectors are quantities with magnitude AND direction, e.g.
displacement,
velocity,
acceleration,
force
For vectors, both magnitude and direction both matter.

4. Differentiate between scalars and vectors
The distance between Dublin and Cork depends on the route you take.
The displacement from Dublin to Cork is constant, has a particular direction, and is different to the displacement from Cork to Dublin.

5. Recognise Quantities as Scalars or Vectors
State whether the following are scalars or vectors:
energy
width
area
weight
current
volume

6. Add Vectors – Triangle Rule
Vectors are represented by arrows.
To add vectors, place the tail of one vector at the head of the other vector. This gives the resultant.
e.g. 𝑎 + 𝑏

7. Add Vectors – Parallelogram Rule
Alternatively, place both vectors with their tails in the same place.
Make each vector one side of a parallelogram.
The diagonal is the resultant vector.

8. Add Vectors
To subtract vectors, turn the subtracted vector around before moving it.
e.g. 3 𝑐 − 𝑑

9. Add Vectors
Copy the following into your copybook and find 𝑎 + 𝑏 for each example:

10. Add Vectors
Copy these diagrams into your copybook and find the following resultants:
2 𝑎 − 𝑏
𝑎 +4 𝑏
1 2 𝑎 𝑏

11. Add Vectors Copy the vectors below into your copy, then add: 𝑎 + 𝑏 + 𝑐
𝑏 + 𝑐 + 𝑎

12. Use Coordinates to Represent Vectors
Vectors are most easily worked with using coordinate geometry.
Instead of the x-y plane, we use the 𝑖 - 𝑗 plane.
𝑖 is one unit long in the i direction.
𝑗 is one unit long in the j direction.

13. Use Coordinates to Work With Vectors
On a single coordinate plane, draw the following vectors:
3 𝑖 +4 𝑗
−3 𝑖 + 𝑗
2 𝑖 −2 𝑗
Calculate the length of each vector.
The i and j parts of a vector are called its components.
𝑖 and 𝑗 are called unit vectors because they are one unit long.

14. Use Coordinates to Work With Vectors
Let 𝑥 =3 𝑖 −2 𝑗 and 𝑦 =4 𝑖 + 𝑗
Plot 𝑥 and 𝑦 on a coordinate plane.
Write 𝑥 + 𝑦 in terms of 𝑖 and 𝑗 .
Find | 𝑥 + 𝑦 |, the magnitude (i.e. length) of 𝑥 + 𝑦 .
Investigate if 𝑥 ⊥ 𝑦 .

15. Decompose Vectors into Components
Given a vector with magnitude 6, making an angle of 30o anticlockwise to the i-axis, find its components.
Find the unit vector in the same direction.

16. Decompose Vectors into Components
Given a vector with magnitude 10, making an angle of 45o clockwise to the i-axis, find its components.
Find the unit vector in the same direction.

17. Calculate the Direction of Vectors
Find the angle that each of the following vectors makes with the i-axis:
𝑖 + 𝑗
4 𝑖 −2 𝑗
2 𝑖 − 𝑗 +4 𝑖 +3 𝑗
2 𝑥 + 𝑦 , where 𝑥 =4 𝑖 +3 𝑗 and 𝑦 =6 𝑖 −8 𝑗

18. Solve Problems about Vectors
Given 𝑎 =4 𝑖 −10 𝑗 and 𝑏 =7 𝑖 +5 𝑗 , find 𝑡 such that 𝑎 +𝑡 𝑏 is a vector pointing along the i-axis (i.e. has no j- component).
If 𝑚 =2 𝑖 − 𝑗 and 𝑛 =4 𝑖 +3 𝑗 , find 𝑘 and 𝑙 such that 𝑘 𝑚 + 𝑙 𝑛 =2 𝑖 −6 𝑗 .
If 11 𝑖 −𝑘 𝑗 =| 𝑖 + 𝑗 |, find two possible values for 𝑘.
Prove that 𝑖 +3 𝑗 ⊥6 𝑖 −2 𝑗 .
If 9 𝑖 −𝑡 𝑗 ⊥2 𝑖 +6 𝑗 , find the value of 𝑡.

19. Solve Problems about Vectors
𝑝 is a vector of magnitude 35 cm; 𝑞 is a vector of magnitude 13 cm. If tan 𝛼 = 4 3 and tan 𝛽 = 5 12 ,
write 𝑝 and 𝑞 in terms of 𝑖 and 𝑗 .
Show that 𝑝 + 𝑞 makes an angle of 45o to the i-axis.