Vectors and Scalars
A SCALAR is ANY quantity in physics that has MAGNITUDE, but NOT a direction associated with it. Magnitude A numerical value with units. Scalar Example Speed Distance Age Heat Number of horses behind the school Magnitude 20 m/s 10 m 15 years 1000 calories I guess: 12
A VECTOR is ANY quantity in physics that has BOTH MAGNITUDE and DIRECTION. Vector Magnitude & Direction Velocity 20 m/s, N Acceleration 10 m/s/s, E Force 5 N, West A picture is worth a thousand word, at least they say so. Vectors are typically illustrated by drawing an ARROW above the symbol. The arrow is used to convey direction and magnitude. v Tail 25 0 Head displacement x = 6 cm, 25 0 length = magnitude 6 cm 25 0 above x-axis = direction
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.
A resultant (the real one) velocity is sometimes the result of combining two or more velocities.
A small plane is heading south at speed of 200 km/h (If there was no wind plane s velocity would be 200 km/h south) 1. The plane encounters a tailwind of 80 km/h. 2. It s Texas: the wind changes direction suddenly 180 0. Velocity vectors are now in opposite direction. 200 km h 80 km h 200 km h 80 km h 280 km h 200 km h 120 km h resulting velocity relative to the ground is 280 km/h Flying against a 80 km/h wind, the plane travels only 120 km in one hour relative to the ground.
You can use common sense to find resulting velocity of the plane in the case of tailwind and headwind, but if the wind changes direction once more and wind velocity is now at different angle, combining velocities is not any more trivial. Then, it s just right time to use vector algebra.
3. The plane encounters a crosswind of 80 km/h. Will the crosswind speed up the plane, slow it down, or have no effect? 200 km h 80 km h 80 km h HELP: In one hour plane will move 80 km east and 200 km south, So it will cover more distance in one hour then if it was moving south only at 200 km/h. To find that out we have to add these two vectors. The sum of these two vectors is called RESULTANT. The magnitude of resultant velocity (speed v) can be found using Pythagorean theorem 2 2 2 2 2 2 v= v 1+v 2= (200km/h) + (80km/h) = 46400km /h 200 km h RESULTANT RESULTANT VECTOR (RESULTANT VELOCITY) v = 215 km/h So relative to the ground, the plane moves 215 km/h southeasterly. Very unusual math, isn t it? You added 200 km/h and 80 km/h and you get 215 km/h. 1 + 1 is not necessarily 2 in vector algebra.
200 km h 280 km h 120 km h 215 km h 180 km h If the air velocity was not at the right angle to the plane velocity, you intuitively know that the speed of the plane would be different. So we are coming to the surprising result. 200 + 80 can be almost anything if 200 and 80 have direction.
Vector Addition: 6 + 5 =? Till now you naively thought that 6 + 5 = 11. Not so fast In vector algebra 6 + 5 can be 10 and 2, and 8, and When two forces are acting on you, for example 5N and 6N, the resultant force, the one that can replace these two having the same effect, will depend on directions of 5N force and 6N force. Adding these two vectors will not necessarily result in a force of 11N. 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. Vectors are quantities which include direction. As such, the addition of two or more vectors must take into account their directions.
There are a number of methods for carrying out the addition of two (or more) vectors. The most common methods are: "head-to-tail" and parallelogram method of vector addition. We ll first do head-to-tail method, but before that, we have to introduce multiplication of vector by scalar.
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 it around. It is determined ONLY by magnitude and direction, NOT by starting point.
Multiplying vector by a scalar Multiplying a vector by a scalar will ONLY CHANGE its magnitude. A 2A 3A 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 A Opposite vectors A - A 3A
Vector addition - head-to-tail method vectors: 6 units,e + 5 units,30 0 examples: v velocity: 6 m/s, E + 5 m/s, 30 0 a acceleration: 6 m/s 2, E + 5 m/s 2, 30 0 F force: 6 N, E + 5 N, 30 0 6 + 5 30 0 you can ONLY add the same kind (apples + apples) 1. Vectors are drawn to scale in given direction. 2. The second vector is then drawn such that its tail is positioned at the head of the first vector. 3. The sum of two such vectors is the third vector which stretches from the tail of the first vector to the head of the second vector. This third vector is known as the "resultant" - it is the result of adding the two vectors. The resultant is the vector sum of the two individual vectors. So, you can see now that magnitude of the resultant is dependent upon the direction which the two individual vectors have.
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. Adding A + B + C + D + E yields the same result as adding C + B + A + D + E or D + E + A + B + C. The resultant, shown as the green vector, has the same magnitude and direction regardless of the order in which the five individual vectors are added.
Example: A man walks 54.5 meters east, then 30 meters west. Calculate his displacement relative to where he started? 24.5 m, E 54.5 m, E 30 m, W + Example: A man walks 54.5 meters east, then again 30 meters east. Calculate his displacement relative to where he started? 54.5 m, E 30 m, E + 84.5 m, E Example: A man walks 54.5 meters east, then 30 meters north. Calculate his displacement relative to where he started? 62.2 m, NE 30 m, N 2 2 54.5 + 30.0 = 62.2 54.5 m, E + The sum 54.5 m + 30 m depends on their directions if they are vectors.
BUT..what about the VALUE of the angle??? Just putting North of East on the answer is NOT specific enough for the direction. We MUST find the VALUE of the angle. 62.2 m, NE 30 m, N 30 θ = arc tan 54.5 θ 54.5 m, E θ = 29 0 So the COMPLETE final answer is : 62.2 m, 29 0 or 62.2 m @ 29 0
A boat moves with a velocity of 15 m/s, N in a river which flows with a velocity of 8.0 m/s, west. Calculate the boat's resultant velocity with respect to due north. v 8.0 m/s, W θ 15 m/s, N 2 2 v = 8 + 15 = 17 m/s 8 tan θ = = 0.5333 15 θ = arctan (0.5333) = [tan 1 (0.5333)] = 28.1 The Final Answer : v v = 17 m/s @ 28.1 = 17 m/s,118.1 W of N
Example A bear, searching for food wanders 35 meters east then 20 meters north. Frustrated, he wanders another 12 meters west then 6 meters south. Calculate the bear's displacement. 2 2 R = 14 +23 = 26.93m 12 m, W 6 m, S θ 20 m, N 35 m, E R 14 m 14 tanθ = = 0.6087 23-1 o θ = tan (0.6087) = 31.3 23 m The Final Answer: RR = 27 mm @ 31 0 = 27 mm, 31 0
Vector addition comparison between head-to-tail and parallelogram method Two methods for vector addition are equivalent. "head-to-tail" method of vector addition parallelogram method of vector addition
"head-to-tail" method of vector addition B B C B + + A parallelogram method of vector addition B A B A A C The resultant vector CC is the vector sum of the two individual vectors. CC = AA + BB
The only difference is that it is much easier to use "head-to-tail" method when you have to add several vectors. What a mess if you try to do it using parallelogram method. At least for me!!!!
Remember the plane with velocities not at right angles to each other. You can find resultant velocity graphically, but now you CANNOT use Pythagorean theorem to get speed. If you drew scaled diagram you can simply use ruler and protractor to find both speed and angle. Or you can use analytical way of adding them. LATER!!! v2 v 1 + v 2 = v v 1 v
SUBTRACTION is adding opposite vector. C = A - B = A + (-B)
I WANT YOU TO DO IT NOW
Components of Vectors Any vector can be resolved into two component vectors. These two vectors are called components. Vertical component y component of the vector A y θ A x A Horizontal component x component of the vector A x = A cos θ A y = A sin θ Vector addition: Sum of two vectors gives resultant vector. A = A x + A y A θ = arc tan A if the vector is in the first quandrant; if not, find θ from the picture. y x
Examle: A plane moves with velocity of 34 m/s @ 48. Calculate the plane's horizontal and vertical velocity components. We could have asked: the plane moves with velocity of 34 A m/s @ 48. It is heading north, but the wind is blowing east. Find the speed of both, plane and wind. v y v θ v x v = 34 m/s @ 48. Find v x and v y v x = 34 m/s cos 48 = 23 m/s v y = 34 m/s sin 48 = 25 m/s wind plane
A plane moves with a velocity of 63.5 m/s at 32 degrees South of East. Calculate the plane's horizontal and vertical velocity components. vv xx = 63.5 cos( 32 0 ) = 53.9 mm/ss V y v x =? 32 0 63.5 m/s V y =? vv yy = 63.5 sin( 32 0 ) = 33.6 mm/ss
If you know x- and y- components of a vector A you can find the magnitude and direction of that vector: Let: F x = 4 N F y = 3 N. Find magnitude (always positive) and direction. F y F θ F x 2 2 F= 4 +3 =5N θ = arc tan (¾) = 37 0 F = 5N @37 0
Vector addition analytically C y B A C B y C = A+ B C = A + B = Acosθ + B cosθ x x x 1 2 C = A + B = Asinθ + B sinθ y y y 1 2 A x A y B x C x
example: F = 68 N@ 24 = 32 N @ 65 1 F 2 F 2 F 1 F F = F + F F 2 1 2 F x = F 1x + F 2x = 68 cos24 0 + 32 cos65 0 = 75.6 N F y = F 1y + F 2y = 68 sin24 0 + 32 sin65 0 = 56.7 N F = F2 + F2 y = 94.5N θ = arc tan (56.7/75.6) = 36.9 0 x F = 94. 5N @ 37 0
vector components and sum simple addition additions