Vectors:Resolution and Addition

Vectors: The Basics

• Vector: has both magnitude and direction– Examples: displacement, velocity, acceleration

• Scalar: has magnitude but no direction– Examples: distance, speed, time, mass

• Drawing Vectors– represented by arrows– length represents magnitude– angle and head of arrow shows direction (angle)– labeled with quantity and/or value

v 1 = 3 m/s

30

Vector Components

• Some vectors point directly along x or y axes, but most do not.

• Components– how much in x & y directions– components are still vectors

have direction– 2 component vectors

represent original vector

Henderson, Tom (n.d.) Vector Components. The Physics Classroom. Retrieved from http://www.physicsclassroom.com/class/vectors/u3l1d.cfm

Vector Resolution: Graphical Method

• Sketch projection lines to x & y axis

• Draw component vectors along axes

Hepburn, Carl Jason. (n.d.) Scalars and Vectors. Splung.com. Retrieved from http://www.splung.com/content/sid/1/page/vectors

Vector Resolution: Algebraic Method

• Visualize vector and its components as a right triangle.• Use SOH CAH TOA to calculate components

Fullerton, Daniel B. (n.d.) Regents Physics Math Review. APlusPhysics.com. Retrieved from http://www.aplusphysics.com/courses/regents/mathreview/regents_math_review.html#

Vector Resolution: An Example• You might want to click here for a video review of trigonometry

and listen to this song about SOH CAH TOA. If you feel comfortable with trigonometry, just click on the link below.If the links do not work, copy & paste into your browser.

• Watch this video demonstration of vector resolution done graphically and algebraically.

dejongescience. (2010, July 11). Basic vector resolution. Video posted to http://www.youtube.com/watch?v=T4tqR4aUh90

Khan, Salman. (n.d.) Basic Trigonometry. Video posted to http://www.khanacademy.org/math/trigonometry/v/basic-trigonometry

KDogde314. (2010, March 25). SOH CAH TOA Song. Video posted to http://www.youtube.com/watch?v=8xwA4DHWLk0

Vector Addition

• Add vectors to find the net (overall) magnitude and quantity of what is being measured (i.e. displacement, velocity, force).

• Resultant: the one vector that represents the sum of two or more vectors

• Only vectors of the same quantity and units may be added or subtracted– You can’t add a velocity vector and a displacement

vector– You can’t add a velocity in m/s with a velocity in mph

Vector Addition: Graphical Method

• Draw 1st vector• Begin tail of 2nd vector at

head of 1st vector• Begin tail of 3rd vector at

head of 2nd vector• Continue until there are

no more vectors• Draw resultant vector

from tail of 1st vector to head of last vector. Label resultant.Appendix: Simple Vectors for Preliminary High School Physics. (2007, April 26) Retrieved from the Bored of Studies

wiki: http://www.boredofstudies.org/wiki/Appendix_-_Simple_Vectors_for_Preliminary_High_School_Physics

Vectors Can Be Added In Any Order• Vectors can be moved parallel to themselves in a

diagram.• Vectors can be added in any order.– Resultant will always be the same.

Henderson, Tom (n.d.) Vector Addition: The Order Does NOT Matter. The Physics Classroom. Retrieved from http://www.physicsclassroom.com/mmedia/vectors/ao.cfm

Resultant of Collinear Vectors

Same direction: add Opposite directions: subtract

Resultant of Perpendicular VectorsVectors: sides of a right triangleResultant: hypotenuseHow to solve?: Pythagorean theoremInverse tangent to find angle

2 21 2

2 23 4

9 16

25

5 cm

s s s

s

s

s

s

What is the turtle’s resultant displacement from Po to Pf?

12 1

1

tan ( / )

tan (4 / 3)

53.1

s s

Resultant: 5 cm @ 53.1 N of E

Vector Addition: Examples

• Let’s make sure you get it.Watch this video demonstration of the addition of 2 perpendicular vectors.

• But, wait!!!What do we do when our vectors are not perpendicular or there are 3 or more vectors?

• Watch this video demonstration to find out!

dejongescience. (2010, July 11). Vector addition at right angles. Video posted to http://www.youtube.com/watch?feature=endscreen&v=i2S9iRLXVqw&NR=1

PhunScience. (2011, August 1). Adding Vectors: How to Find the Resultant of Three or More Vectors. Video posted to http://www.youtube.com/watch?v=g_TnqKX5ybY&feature=related

• Use the “vector components” method by first resolving all vectors into their components.

• Magnitude:– Find the sum of all x-components and all y-components.– These are the component s of the resultant vector.

– Use Pythagorean Theorem to find resultant from components.

– Use Inverse tangent to find precise direction of resultant.

Summary: Vector Addition of Non-Perpendicular Vectors (or 3 or more vectors)

xxxx CBAR

2 2x yR R R

yyyy CBAR

1tan ( )y

x

R

R