Introduction to vectors

Transcription

1 Lecture 4 Introduction to vectors Course website: Lecture Capture: , Fall 2013, Lecture 3

2 Outline Chapter 3. Section ectors and Scalars Addition of ectors Subtraction of ectors Multiplication of a ector b a Scalar Adding ectors b Components Unit ectors

3 ector and Scalars ector quantit has both direction and magnitude e.g. displacement, velocit, acceleration, force, momentum r,v,a r Scalar quantit has onl magnitude (no need in direction) e.g. distance, speed, temperature, mass, time, densit, volume

4 Addition of ectors (1D) For vectors in one dimension, simple addition and subtraction are all that is needed. Eas!!!! If the vectors are in the same direction If the vectors are in opposite directions

5 Addition of ectors (2D). Graphical Methods Triangle method. If the motion is in two dimensions, the situation is somewhat more complicated. Tail-to-Tip method Draw first vector Draw second vector, placing tail at tip of first vector Arrow from tail of 1 st vector to tip of 2 nd vector: resultant A B

6 Addition of ectors (2D). Graphical Methods Parallelogram method. Parallelogram method The two vectors, A and B, are drawn as the sides of the parallelogram and the resultant, C = A + B, is its diagonal B A Commutative propert of vectors C = A + B = B + A

7 B Subtraction of vectors A is a vector with the same magnitude as but in the opposite direction. So we can rewrite subtraction as addition A B B A So, we add the negative vector. = ( B) B B A B C = A B

8 ConcepTest 1 ector Addition You are adding vectors of length 20 and 40 units. What is the onl possible resultant magnitude that ou can obtain out of the following choices? A) 0 B) 18 C) 37 D) 64 E) 100

9 ConcepTest 1 You are adding vectors of length 20 and 40 units. What is the onl possible resultant magnitude that ou can obtain out of the following choices? ector Addition A) 0 B) 18 C) 37 D) 64 E) The minimum resultant occurs when the vectors are opposite, giving 20 units. The maximum resultant occurs when the vectors are aligned, giving 60 units. Anthing in between is also possible for angles Min=40-20=20 40 Max=40+20=60 Resultant is between 20 and 60 between 0 and 180.

10 Multiplication of a ector b a Scalar A A vector can be multiplied b a scalar b(positive); the result is a vector B that has the same direction but a magnitude. ba If b is negative, the resultant vector points in the opposite direction. B = 1.5 A C = 2.0 A A

11 Addition of three or more vectors Can use tip to tail for more than 2 vectors D = A + B + C A + + = B C A B C Order of addition does not matter

12 ector components It is customar to resolve a vector into components along mutuall perpendicular directions.

13 Determining vector components Given and θ, we can find x, In 2D, we can alwas write an vector as the sum of a vector in the x-direction, and one in the -direction. Given, θ (magnitude, direction), we can find x and = x + x cos so x cos sin so sin q x

14 Given x and, we can find, θ x, are the legs of the right triangle and are therefore perpendicular ector as the hpotenuse. So, the magnitudes of the vectors satisf the Pthagorean Theorem. so 2 2 x tan 2 x tan x 1 2 x 2 q x x

15 Example 1 A vector is given b its magnitude and direction (,) What is the x, -component of the vector? 10m 30 above x axis

16 Example 2 A vector is given b its vector components: x 2, Write the vector in terms of magnitude and direction. -2 x x 4 4 magnitude 2 x x tan tan 2 tan from x axis from x axis

17 Adding vectors b components x, x, 2 x, Given and, how can we find? = x 2x, Adding corresponding components 1x x = 1x + 2x 2x x

18 Unit ectors (1) As we said before, a vector has both magnitude and direction. Now, it s time to simplif a notation of direction: Let s introduce unit vectors iˆ, ˆ, j kˆ known as unit vectors z Unit vectors have a magnitude of 1 i = j = k =1 k i x The point along major axes of our coordinate sstem j

19 Unit ectors (2) Writing a vector with unit vectors is equivalent to multipling each unit vector b a scalar x iˆ ; x iˆ x ˆj ˆj ( x, ) j i x xˆ i x If a vector has components: x 4, In unit vector notation, we write 4iˆ 3 3 ˆj (4,3)

20 Example: ector Addition/Subtraction A hiker traces her movement along a trail. The first leg is a flat hike to the foot of the mountain: On the second leg, she climbs the mountain: D 1 = (2500m)î +(500m)ĵ D 2 = (500m)î +(700m)ĵ +(700m) ˆk On the third, she walks along a plateau: Then she falls off a cliff: What is the hiker s final displacement? D = D 1 + D 2 + D 3 + D 4 D = (3000m)î D 3 = (600m)ĵ D 4 = -(500m) ˆk = (2500m)î + (500m) ĵ + (500m)î + (700m) ĵ + (700m) ˆk + (600m) ĵ - (500m) ˆk +(1800m)ĵ +(200m) ˆk

21 Relative elocit So far we have just added/subtracted displacement vectors Ma find situations to add or subtract other tpes of vectors, sa velocit vectors Can onl add or subtract the same tpe of vectors

22 Relative elocit + A train moves at 25 m/s relative to the ground Your velocit relative to the train is 5 m/s So our velocit relative to the ground is: 20 m/s - 5m/s m/s 5m/s= m/s x

23 Boat in the river. Derivation of the relative velocit equation 3-15 BS = BW + WS BW means velocit of the boat relative to water B W S Boat Water Shore

24 Relative elocit - 2D BS River current WS =1.20m/s θ BW B W S =1.85m/s Boat Water Shore BW means velocit of the boat relative to water A boat s speed in still water is 1.85m/s. The river flows with a 1.20m/s current. If we want to directl cross the stream at what upstream angle (see diagram) should the boat be pointed? BS = BW + WS opposite sin hpotenuse 1.20 sin sin 1 (0.6486) 40.4

25 Summar ectors Graphical Methods Addition and Subtraction Multiplication b a scalar Components Unit vectors Displacement & velocit vectors Relative elocit

26 Thank ou See ou on Wednesda

27 ConcepTest 1 ectors I If two vectors are given such that A + B = 0, what can ou sa about the magnitude and direction of vectors A and B? A) same magnitude, but can be in an direction B) same magnitude, but must be in the same direction C) different magnitudes, but must be in the same direction D) same magnitude, but must be in opposite directions E) different magnitudes, but must be in opposite directions

28 ConcepTest 1 ectors I If two vectors are given such that A + B = 0, what can ou sa about the magnitude and direction of vectors A and B? A) same magnitude, but can be in an direction B) same magnitude, but must be in the same direction C) different magnitudes, but must be in the sam direction D) same magnitude, but must be in opposite directions E) different magnitudes, but must be in opposite directions B A The magnitudes must be the same, but one vector must be pointing in the opposite direction of the other in order for the sum to come out to zero. You can prove this with the tip-to-tail method.

29 ConcepTest 2 ectors II Pthagorean Theorem Given that A + B = C, and that lal 2 + lbl 2 = lcl 2, how are vectors A and B oriented with respect to each other? 1) the are perpendicular to each other 2) the are parallel and in the same direction 3) the are parallel but in the opposite direction 4) the are at 45 to each other 5) the can be at an angle to each other A B Note that the magnitudes of the vectors satisf the Pthagorean Theorem. This suggests that the form a right triangle, with vector C as the hpotenuse. Thus, A and B are the legs of the right triangle and are therefore perpendicular.