Algebraic Addition of Vectors Now you shall cast aside your rulers and protractors, and use your head and calculators to add vectors…

Adding Vectors in One Dimension + 2 N = 6 N This is so easy!!! When do I need a calculator??

Adding Vectors In Two Dimensions If you’re really lucky, the vectors might be perpendicular to each other: A B C Where A and B are the magnitudes of A and B, and C is the magnitude of C. 

What If You’re Not So Lucky? Well, the process is a little more complicated. First we need to talk about components of vectors. Any vector may be considered the vector sum of an infinite number of other vector combinations:

Components of Vectors Any vector may be expressed as the sum of a vector parallel to the horizontal axis (Ax), and a vector parallel to the vertical axis (Ay): A Ay Ax

How Can We Find These? Since Ax is perpendicular to Ay the following is true: A Ay Ax 

What Good are These Components? Cx Cy A C B By Bx Ay Ax

Steps for Adding Vectors… … That aren’t parallel or perpendicular to each other: Break all vectors into their horizontal and vertical components. Add the horizontal components together to get the horizontal component of the resultant. Add the vertical components together to get the vertical component of the resultant. Add the components of the resultant using the Pythagorean Theorem. Use the components of the resultant and a trigonometric function to get the direction of the resultant.

For Example: Let’s Add Vectors A and B:   A B A = 13 N @ 22.6 B = 5 N @ 126.9

(1) Break all vectors into their horizontal and vertical components. B = 5 N @ 126.9 Bx= 5 N cos 126.9  = 3 N By = 5 N sin 126.9 = 4 N Ay Ax A A = 13 N @ 22.6 Ax= 13 N cos 22.6  = 12 N Ay = 13 N sin 22.6 = 5 N By Bx B

(2) And (3) Add the Components Ax= 13 N cos 22.6  = 12 N Bx= 5 N cos 126.9  = 3 N Ax + Bx= Cx = 9 N Ay = 13 N sin 22.6 = 5 N By = 5 N sin 126.9 = 4 N Ay + By= Cy = 9 N

(4) Add the Components for the Resultant Cx = Cy = C C2 = Cx2 + Cy2 C2 = (9 N)2 + (9 N)2 C = 12.7 N The components will ALWAYS be perpendicular to each other, so we’ll always use the Pythagorean Theorem

(5) Find the Angle 9 N  C Cy = Cx = You know the legs of the right triangle, as well as the hypotenuse, so you can use any trig function…

Find the Vector Sum of A + B C= 12.7 N @ 45 C B A

Hmm…

A Word About Directions Vector Components

A Few More Words… A Ay Ax  A = 5 m/s @  = 150 Ax= 5 m/s cos 150 = - 4.33 m/s OR Ax = 4.33 m/s

A Few More Words… A Ay  Ax A = 5 m/s @  = 30 Ax= 5 m/s cos 30 = 4.33 m/s Which, by inspection, you can call 4.33 m/s