Adding Vectors Vectors are ‘magnitudes’(ie: values) with a direction The speed and direction of a car is a vector. The strength and direction that you push something with a force is a vector. Your ‘displacement’ if you walk some city blocks is a vector. Vectors can be added. If you take Physics you will need to be rather proficient with vectors!

Two Displacement Vectors Vector A: Walking three blocks north 3 Vector B: Walking four blocks east. 4 ‘Tail’ of the vector ‘Head’ of the vector

Adding Vectors: Tail to Head Method Put the ‘tail’ of Vector B to the ‘head’ of Vector A. 4 3 Vector C is the result of adding Vector A to Vector B. Vector C is called the ‘resultant’. Vector C has a length and a direction.

Show adding again 4 3 So the resultant Vector C is found by adding the tail of Vector B to the head of Vector A. Vector C has a length and a direction; but what is that length and direction?

Find Length and Direction of Resultant 3 4 Just measure the length of C with a ruler and its angle with a protractor 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 180 160 170 1 2 3 4 5 The resultant Vector C is 5 long in a direction 53 degrees to the right of Vector A. We have measured the length and the direction of Vector C.

Resultant Velocity: Swimming ‘Velocity’ is a speed in a given direction Sheena is swimming partly ‘upstream’ and across a river at 4 km/h in a direction 45° from the shore. The current is at 2 km/h and is parallel to the river of course. Find the actual speed and direction that Sheena is actually moving by adding the two vectors. Or you could say Sheena’s direction is 45° + 28° = 73° from the shore Current 4 cm for 2 km/h Current 4 cm for 2 km/h Sheena 8 cm for 4 km/h 6 cm for 3 km/h 28° Scale: A length of 2 cm is 1 km/h 45° Shoreline Measure the length of the resultant and its angle. Sheena is swimming at 3 km/h at an angle of 28 degrees to the left of her swimming velocity

Vectors: Using More Tools The previous examples were conceptual; they gave you the idea. Let us be a bit more rigorous in our method of adding vectors. It works out better if we are drawing vectors if we use a ‘grid’ and a certain ‘scale’ of measurement. The grid will help us measure the angles more easily; the scale will allow us to make vectors the right length for different situations. And it will help if we agree on a better way to measure angles. We need an angle that is the ‘zero’ angle, and then measure everything from that. [000°] [090°] [180°] [270°] [135°] North East West South The bearing navigation reference for measuring angles: A ‘grid’:

Vectors: Motion Example Have you ever noticed boat or airplane motion? The boat or airplane will be pointing (‘heading’) in a certain direction with a certain speed, but actually traveling in a different direction (its ‘course’) and speed because the air itself is moving (moving air is called ‘wind’!) The direction the aircraft is pointing is called its ‘heading’ and its speed through the air is called ‘airspeed’. Its actual speed relative to the ground is called its ‘ground speed’; its actual direction is called its ‘track’. It is the same idea with boats in a current wind no wind wind

Aircraft Velocity Example An aircraft is flying North [000°] at 200 mph. The wind is from the west [270°] at 80 mph. What is the actual resultant motion vector of the airplane? 1. Establish a scale. Lets say that one cm is 20 mph. 2. Draw the aircraft vector from some convenient point. Call it vector A 4cm = 80mph The resultant is 10.8 cm long, so 216 mph The aircraft is really moving in a direction [022°] at 216 mph. 3. Add the wind vector to the head of the aircraft vector. Call it W. 10 cm for 200 mph The angle is [022°] 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 4. Measure the length and direction of the resultant Vector R. Scale: 1 cm = 20 mph

Displacement Vectors: Water Balloon Terrance and Monique are playing with water balloons. They are initially standing together. Monique runs 8 metres from Terrance in direction [045°] then 6 metres in direction [120°]. How far must Terrance throw his water balloon and in which direction to get Monique? Add Monique’s two displacements together by adding her two vectors. 6 meters@[120°] 8 meters@[045°] Scale: 1 cm = 1 metre 11.2 cm or 11.2 meters@[076°] Terrance must throw his balloon in the direction [076°] and a distance of 11.2 meters to bean Monique 1cm = 1m Displacement just means ‘change in position’

Properties of Vectors: Commutative Law You know that 2 + 4 = 6 and 4 + 2 = 6. This is called the commutative law of addition. The order in which you add numbers does not matter. You also know that the commutative law works with matrices The commutative law works for vectors too! Going two blocks east and three blocks south is the same as going three blocks south and two blocks east!

Commutative Law of Vectors + = + =

Subtracting Vectors Can you subtract vectors? Of course! Just add an ‘opposite’ vector! Just like 5 – 3 = 5 + (–3) You will not subtract Vectors in Grade 12 Math, but you will in Physics!

Multiply Vectors by a Scalar Here is Vector A What is 2*Vector A? It is really just Vector A + Vector A (that is what multiplying by two is!) +

Multiply a Vector by a Vector Can you multiply a Vector by a Vector? Yes, you can! But that is more of a university lesson, so we will not do it here in Grade 12! When you multiply Vector X by Vector Y you end up leaving the paper in a third dimension; so save that idea for university!

Adding Vectors: Parallelogram Method There is another way to add vectors graphically! It is a bit more intuitive for some people. The parallelogram method! All you do is transfer the directions of the vectors to the head of each vector and where they cross is the resultant

Parallelogram Method Example Rick and CJ are trying to pull out a tree stump. Rick is pulling with a force of 100 lbs in the direction N45W, CJ with a force of 160 lbs N45E. What is the resultant force on the tree stump? Measure 9cm or 180 lbs in a direction 34° to the left of CJ or N11W Resultant force Just transfer the direction of each vector to the head of the other vector 34° CJ Rick 160 100 20 lbs = 1 cm

Vector Addition Using Trigonometry So do you walk around with graph paper and a ruler and a protractor? Even I do not do that! So lets figure out how to do vectors using trigonometry, at least you only need a calculator for that! Have you noticed that adding two vectors has actually been using a triangle?

Adding Vectors using Trigonometry Caution! Lengths are not to scale now, we can just use trig, so we don’t have to accurately draw vectors. We can just sketch them Opposite 6 B From Pythagoras we know that R2 = A2 + B2 8 A Adjacent R So R2 = 82 + 62 So R is 10 long if you do the calculation  But what is the angle ?

Cosine and Sine Law The last example had a 90° corner. But not all triangles and vectors have a 90° corner. Pythagoras does not work if there is no 90 ° angle! So we will need to use our Cosine and Sine laws from Grade 10 if the triangle is not a right triangle! a b c A B C And you will need to remember some other basic rules from Grade 10 Geometry!

Vectors using Sine and Cosine Law An aircraft is flying on a heading of [045°] at 300 mph through the air. The wind is from a direction of [300°] at 60 mph. What is the actual vector motion of the aircraft? (track and ground speed) [000°] [045°] The aircraft is flying at 321 mph in a direction [055°] Caution, we are just sketching our vectors here. The directions and lengths may not be to scale or accurate! 75° 60 105° W 300 A [120°]  R Therefore:  = 10.4°. So R is in direction [045 °] + 10.4 ° = [055.4 °]

Example 2: Sine and Cosine Law Nathan is out snowmobiling and gets lost. He knew he went 6 km north [000°] then he turned 60° to the right and went another 4 km. How far is he from home and in which direction? Cosine Law: 6 4 60° So R = 8.7 km Sine Law: 120° 8.7 km  So Nathan is 8.7 km from home on a bearing of [023.5°]

Protractor 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 180 160 170 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 1 2 3 4 5