Starter  

Extra question  

What are scalars and vectors?

Scalars & Vectors Lesson Goals: Be able to identify scalar and vector quantities Be able to determine a resultant vector using the parallelogram or triangle rule Be able to resolve a vector into perpendicular components

Khan Academy https://www.khanacademy.org/math/precalculus/vectors-precalc One resource you can use outside the classroom to help consolidate work. Also:

Antonine Education http://www.antonine-education.co.uk/index.htm

Scalar Quantity – A quantity which has a magnitude but not a direction Vector Quantity – A quantity with both magnitude and direction

Scalar or Vector? Distance Mass Velocity Displacement Energy Force Acceleration Speed Density Power Current Momentum Frequency Scalar or Vector?

Parallel Vectors If they point in the same direction, add them If they point in opposite directions, add them with one direction taken to be negative (normally left).

Taking into account their direction How do we combine scalar quantities? Add them together How do we combine vector quantities? Carefully! Taking into account their direction

Perpendicular Vectors – The Triangle Rule This is a geometric construction that can be used to determine the resultant of two vectors. In this construction the arrows must be drawn tip to tail. If right-angles triangle then use Pythagoras R b a Resultant Vector R2 = a2 + b2

Trigonometry recap - V.Important!!! sin θ = opposite ÷ hypotenuse cos θ = adjacent ÷ hypotenuse tan θ = opposite ÷ adjacent

Calculate the missing side or angle ? ? 3 B 6 30° Calculate the missing side or angle 4 7 ? C 9 15° ? D 10

What about when the vectors are at a random angle?

Resolving Vectors into Perpendicular Components F θ Horizontal component, Fx = F cos θ Vertical component, Fy = F sin θ

Resolve the following forces into perpendicular components B A 8N 4N 30° 50° D C 17N 12N 40° 15°

How do we combine two or more non-perpendicular vectors ?

The Parallelogram Rule This is a geometric construction that can be used to determine the resultant of two vectors E.g. What is the resultant of the following? 8N B A 5N 45° 20°

But - Let’s just use the triangle rule from now on! What is the resultant magnitude and direction of the following? 8N B A 5N 45° 20°

How would you tackle this??? 8N B 3N A 70° 80° 60° 70° 50° 30° D C 7N 5N

What is the resultant force? Step 1 Draw resolved vector arrows. Calculate the horizontal and vertical components of each vector 60° 50° 70° 80° 3N 8N 5N 7N 30° B A D C

A What is the resultant force? Horizontally: - 3 cos 10 = - 2.95N B 60° 50° 70° 80° 3N 8N 5N 7N 30° A Horizontally: - 3 cos 10 = - 2.95N Vertically: 3 sin 10 = 0.52N D C

B What is the resultant force? Horizontally: 8 cos 20 = 7.52N 60° 50° 70° 80° 3N 8N 5N 7N 30° A Horizontally: 8 cos 20 = 7.52N Vertically: 8 sin 20 = 2.74N D C

C What is the resultant force? Horizontally: 7 cos 40 = 5.36N 60° 50° 70° 80° 3N 8N 5N 7N 30° B Horizontally: 7 cos 40 = 5.36N Vertically: -7 sin 40 = -4.50N A D C

D What is the resultant force? Horizontally: -5 cos 60 = -2.5N 60° 50° 70° 80° 3N 8N 5N 7N 30° B Horizontally: -5 cos 60 = -2.5N Vertically: -5 sin 60 = -4.33N A D C

What is the resultant force? B 60° 50° 70° 80° 3N 8N 5N 7N 30° A Step 2 Add horizontal and vertical components D C

What is the resultant force? 60° 50° 70° 80° 3N 8N 5N 7N 30° B Horizontally: -2.95 + 7.52 + 5.36 – 2.5 = 7.43N Vertically: 0.52 + 2.74 – 4.50 – 4.33 = - 5.57N A C D

Find the resultant of this pair of perpendicular vectors Step 3 Find the resultant of this pair of perpendicular vectors 7.43N 5.57N We could use scale diagrams… I vote for Pythagoras

Tan θ = opp / adj R2 = 7.432 + 5.572 = 5.57 / 7.43 = 86.23 θ = 36.86° R = 9.29N