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Vectors

Vectors. Vectors. 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. Vector Components. The magnitude of the vector is expressed as the length of the arrow.

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Vectors

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  1. Vectors

  2. Vectors • 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

  13. The Plane & the Wind

  14. The Plane & the Wind • If we look at the situation as a right triangle we can use a familiar formula to solve for the resultant – a2 + b2 = c2 – the pythagorean theorem. • C2 = 1002 + 252 • 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 c2 = a2 + b2 • 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

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