1. Vectors

2.  A Vector is a physical measurement that has both magnitude and direction.  Vectors include displacement, velocity, acceleration, and force.  Vectors are usually represented by an arrow.

3. Vector Components  The magnitude of the vector is expressed as the length of the arrow.  The longer the arrow, the larger the magnitude.  All vectors have two components – the head and the tail.

4. Vector Direction  The direction of the vector is usually expressed in degrees.  If we picture the vector on an x-y coordinate plane, the direction takes on more meaning.

5. Vector Addition & Subtraction  Vectors can be added and subtracted to each other.  This is easy if the vectors are facing the exact same direction (addition) or the exact opposite direction (subtraction).

6. Vector Addition  What happens when the vectors aren’t facing the same or opposite direction?  Another method must be used to determine the answer, or resultant.

7. Graphical Method  One method involves placing the vectors tip to tail.  A straight line can then be drawn from the beginning point to the end point.  This line is the resultant of the vectors

8. Order Doesn’t Matter  It doesn’t matter what order we draw the individual vectors.  When we get to the last vector, the resultant will always be equal.  This method has some drawbacks. A measuring device must then be used to calculate the length of the resultant.

9. Vector Components  If we use our knowledge of trigonometry, we know that any diagonal line can be broken down into two components – an x and a y.  We do this by forming a right triangle, with the diagonal line becoming the hypotenuse.

10. Vector Components  If we use the analogy of a puppy pulling on his chain, we can see that the force the puppy is exerting on the chain could be thought of as the hypotenuse of a right triangle.  Therefore we can find the “legs” of the triangle (the components of the vector) and determine how much force the puppy is exerting in both the x (left-right) direction, and the y(up- down) direction

11. The Plane & the Wind  Our analogy can be applied to many real-world situations.  In the animation to the right, we look at the effect of a tailwind, a headwind, and a crosswind on the path of an airplane

12. The Plane & the Wind

14.  If we look at the situation as a right triangle we can use a familiar formula to solve for the resultant – a 2 + b 2 = c 2 – the pythagorean theorem.  C 2 = 100 2 + 25 2  C = 103.078 km/hr

15. The Plane & the Wind tan (θ) = (opposite/adjacent) tan (θ) = (25/100) θ = tan -1 (25/100) θ = 14.036º 103.1 km/hr at 14.036º W of S or… 103.1 km/hr at 75.964º S of W or 103.1 km/hr at 255.964°

16. Boat In a Current  A motor boat traveling 4 m/s, East encounters a current traveling 3.0 m/s, North.  1. What is the resultant velocity of the motor boat?  2. If the width of the river is 80 meters wide, then how much time does it take the boat to travel shore to shore?  3. What distance downstream does the boat reach the opposite shore?

17. Boat In a Current  We could find the resultant speed of the boat by using c 2 = a 2 + b 2  C = 5m/s

18. Boat In a Current  Next we use the horizontal velocity of 4m/s and the horizontal distance of 80m to find the time.  T = 20 seconds (80m / 20m/s)

19. Boat in a Current  Next we use the time we found in #2, t = 20 and use the vertical velocity 3m/s to find how far downstream the current moves the boat.  D = (20sec)(3m/sec) = 60 m