The Science of Vectors Magnitude & Direction

What are they? When we measure things in Science - we not only must know how much (magnitude) but in what direction. This is important - Direction Matters!!

To add two (or more) vectors together graphically using the head-to-tail method you simply draw the first vector anywhere you wish, then draw the second vector with its tail at the head of the first vector. If there are more vectors to be added draw each one with its tail at the head of the preceding vector.

When you are finished drawing all the vectors you should have a chain of vectors, with the tail of each vector (except the first) coincident with the head of the preceding vector. The sum or resultant is a vector drawn from the tail of the first vector in the chain to the head of the last vector in the chain.

Vector addition is commutative; that is, it does not matter in which order you add them. You need only insure that the length and direction of each vector is maintained. Problem: While sorting through his basement, Mr. C happens on a treasure note. The note leads Mr. C. on the following path:

From his house, he goes 100 km due east, turns, and then goes 500 km due north. The note next reads: Turn 45º west of north (northwest) and go 100 km, then turn and go 350 km due north. Evidence that he is getting close to the treasure is the fact that the remaining instructions are written in gold letters on the bark of a tree: Turn 90º west and go 300 km, then turn 45º south of west (southwest) and go 200 km. At long last, the end of the rainbow is in sight….

1.What distance has Mr.C. traveled so far? 2. How far away and in what direction is Mr.C. from home?- (what is his displacement from the origin?)

10 km due east 50 km due north. 10 km 45º west of north 35 km due north 30 km due west 1.What is the total distance traveled? 2.What is the displacement from the origin? (include magnitude & direction)

5N 30  S of E N Move 1 st vector here 5N S 60  5N S

5N 30 degrees S of E N Resultant! 59 degrees S of E Equilibrant 59 degrees N of W 5N S Another name… Parallelogram Method

20N 37  S of W  N 9N N 127  total  180  – 127  = 53  53  

N 9N N 53   9N N 20N 37  S of W RESULTANT S  of W W  of S Equilibrant N  of E E  of N

5m 60  E of S or S 60  E N 5m S 60  HEAD-TO-TAIL METHOD OF VECTOR ADDITION (usually displacement vectors “adding” together to outline a journey) RESULTANT DISPLACEMENT 8.8m 30  E of S or S 30  E 30 

N 60   5N S 5N 60  E of S RESULTANT  E of S 60   5N S 30    30  EQUILIBRANT  W of N PARALLELOGRAM METHOD OF VECTOR ADDITION (usually force vectors acting on a single point)

N 10 N W 16 N 45  S of E 16 N S 45  E 15 N 35  E of N Or 15 N N 35  E 24 N 6  S of W 24 N S 6  W 100  Modified Parallelogram Method For Adding 3 Vectors 1.Draw all 3 vectors concurrently (from a point) 2. Sum 2 of the vectors – “slide” one to add it to the other 35  100 

N 10 N W 24 N 6  S of W 14N 8  S of W Modified Parallelogram Method For Adding 3 Vectors 3.Now use the resultant from the first 2 vectors as a “new” vector to add to the 3 rd vector 4. Sum the vectors – “slide” one to add it to the other 5. Obtain your FINAL resultant!

º N of E 300N E  400N N Force Vector Practice 400N N

Vector Resolution The method of employing trigonometric functions to determine the components of a vector are as follows: 1.construct a sketch (no scale needed) of the vector in the indicated direction; label its magnitude and the angle which it makes with the horizontal. 2.draw a rectangle about the vector such that the vector is the diagonal of the rectangle; beginning at the tail of the vector, sketch vertical and horizontal lines; then sketch horizontal and vertical lines at the head of the vector; the sketched lines will meet to form a parallelogram.tailhead

Vector Resolution 3.draw the components of the vector; the components are the sides of the rectangle; be sure to place arrowheads on these components to indicate their direction (up, down, left, right). 4.meaningfully label the components of the vectors with symbols to indicate which component is being represented by which side; a northward force component would be labeled F-north; a rightward force velocity component might be labeled v-x; etc.

Vector Resolution 5.to determine the length of the side opposite the indicated angle, use the sine function; substitute the magnitude of the vector for the length of the hypotenuse; use some algebra to solve the equation for the length of the side opposite the indicated angle. 6.repeat the above step using the cosine function to determine the length of the side adjacent to the indicated angle.

On a separate sheet – solve the following problem – SHOW ALL WORK- then hand in: Use mathematical methods to determine both the horizontal & vertical components of the following force vector: #1: º with the x axis #2: º with the x axis

º with the x axis A x = A cos  A x = 350N (cos 56º) A x comp = 195.7N  A y = A sin  A y = 350N (sin 56º) A y comp = 290.2N

º with the x axis A x = A cos  A x = 425N (cos 47º) A x comp = 289.9N  A y = A sin  A y = 425N (sin 47º) A y comp = 310.8N

a 2 + b 2 = c 2 (48N) 2 + (27N) 2 = c N 2 = c N = c A x = 48N  A y = 27N  = tan -1 (27N/48N)  = 29.4º

Weight = 50N   = 40º  W y = A sin  W y = 50N (sin 40º) W y comp = 32.1N W x = A cos  W x = 50N (cos 40º) W x comp = 38.3N