Vectors Ch 3

Vectors Vectors are arrows Vectors are arrows They have both size and direction (magnitude & direction – OH YEAH!) They have both size and direction (magnitude & direction – OH YEAH!) Vectors have NO PLACE!!! Vectors have NO PLACE!!!

Scalars Scalars are line segments (no direction) Scalars are line segments (no direction) Example: 24 cookies, 58 hours, etc… Example: 24 cookies, 58 hours, etc…

Vector Examples : Velocity 24 m/s Speed is a scalar Why? No direction

Force 980 N

Combining Vectors If two or more vectors are acting on an object, it is possible to combine (add) them as long as they are drawn head- to -tail head-tail

Vectors in the Same Direction When vectors are in the same direction, their magnitudes combine and the overall value increases. For example…

10 m/s 7 m/s 17 m/s + = Resultant vector

Vectors in Opposite Directions When vectors are in the opposite directions, their magnitudes combine and the overall value decreases. The resultant vector is in the direction of the larger vector

16 m/s10 m/s 6 m/s + = Resultant vector

More than two It is possible to combine more than two vectors It is possible to combine more than two vectors Think of a tug of war….. Think of a tug of war…..

If all of these people are pulling, Then their strengths will combine

7 N 12 N 6 N 4 N

For a total strength of 29 N

It is possible to combine vectors that are moving in directions at angles to each other.

12 N 9 N For example, what would be the effect of these two vectors acting simultaneously? (still draw head-tail)

12 N 9 N ? From the Pythagorean theorem c 2 =a 2 +b 2 the length of the resultant is sq rt( )=15

Remember to draw head-to-tail !!!

12 N 9 N Either way is ok 12 N 9 N 15 N

12 N 9 N 15 N Right angles are always easier to work with 12 N 21 m With the proper technique any combination of vectors can be resolved

Vector Terminology 1) Components: the two vectors (usually a horizontal and a vertical) that can be used to replace a larger vector. * resolution: the process of finding the two components of a vector. * resolution: the process of finding the two components of a vector. 2) Resultant: the sum of at least two other vectors. 3) Equilibrium: the condition established when the resultant of the forces is zero and Newton’s 1 st Law is observed

Basic Trigonometry Functions sin  = opposite/hypotenuse soh cos  = adjacent/hypotenusecah tan  = opposite/adjacenttoa sin  = opposite/hypotenuse soh cos  = adjacent/hypotenusecah tan  = opposite/adjacenttoa hypotenuse  adjacent opposite

Example using Pythagorean Theorem and Tan function An archaeologist climbs the Great Pyramid in Giza, Egypt. If the pyramid’s height is 136m and its width is 2.3x10 2 m, what is the magnitude and the direction of the archaeologist’s displacement while climbing from the bottom of the pyramid to the top? An archaeologist climbs the Great Pyramid in Giza, Egypt. If the pyramid’s height is 136m and its width is 2.3x10 2 m, what is the magnitude and the direction of the archaeologist’s displacement while climbing from the bottom of the pyramid to the top?

YOU TRY PG. 91 # 1-4 YOU TRY PG. 91 # 1-4

Example using Sine and Cosine functions Let’s examine a scene from a new action movie. For this scene a biplane travels at 95 km/hr at an angle of 20 degrees relative to the ground. Attempting to film the plane from below, a camera team travels in a truck, keeping the truck beneath the plane at all times. How fast must the truck travel to remain directly below the plane? Let’s examine a scene from a new action movie. For this scene a biplane travels at 95 km/hr at an angle of 20 degrees relative to the ground. Attempting to film the plane from below, a camera team travels in a truck, keeping the truck beneath the plane at all times. How fast must the truck travel to remain directly below the plane?

YOU TRY PG. 94 # 1-7 YOU TRY PG. 94 # 1-7

Solving Vector Problems 1) Graphically (parallelogram method): a ruler and protractor are used to draw all vectors to scale from head-tail. The answer is the resultant which is drawn from the tail of the first vector to the head of the last vector. r

2) Trigonometrically a) using trig. functions resolve each vector into its horizontal and vertical components. b) draw a “super-sized” right triangle from all of the combined horizontal and vertical components. c) use the Pythagorean theorem to determine the resultant (hypotenuse) AND trig. functions to determine the angle of the resultant.

r The result is the same!! r horizontal vertical c 2 =a 2 +b 2

Example of adding vectors that are not perpendicular A hiker walks 25.5 km from her base camp at 35 degrees south of east. On the second day, she walks 41.0 km in a direction 65 degrees north of east, at which point she discovers a forest ranger’s tower. Determine the magnitude and direction of her resultant displacement between the base camp and the ranger’s tower. A hiker walks 25.5 km from her base camp at 35 degrees south of east. On the second day, she walks 41.0 km in a direction 65 degrees north of east, at which point she discovers a forest ranger’s tower. Determine the magnitude and direction of her resultant displacement between the base camp and the ranger’s tower.

YOU TRY PG. 97 # 1-4 YOU TRY PG. 97 # 1-4