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Vectors – An Overview of Notation, Anatomy, Operations, and a lil’ bit of Physics

Vectors – An Overview of Notation, Anatomy, Operations, and a lil’ bit of Physics Vector – A directed line segment that represents any quantity that has magnitude and direction. Note the initial point comes first in the notation May also let.

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Vectors – An Overview of Notation, Anatomy, Operations, and a lil’ bit of Physics

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  1. Vectors – An Overview of Notation, Anatomy, Operations, and a lil’ bit of Physics Vector – A directed line segment that represents any quantity that has magnitude and direction.

  2. Note the initial point comes first in the notation May also let

  3. Equivalent Vectors – Same length and direction. Important when we want to move vectors representing quantities to more convenient spots for easier usability. Same direction means same slope. Same magnitude means same length.

  4. So much easier to work with vectors whose initial points are the origin. We call these vectors: COMPONENT VECTORS Check out the notation.

  5. Not every vector is in component form! The audacity of some vectors!!! But we can move vectors as long as we maintain their slope and direction (equivalence). - Picture sliding a vector!!

  6. Changing a Vector to Component Form Subtract the coordinates of the initial point from the coordinates of the terminal point. WHY DOES THIS MAKE GEOMETRIC SENSE?

  7. Component vectors are also easier to find the magnitude of!!

  8. 1.) Sketch the vector with init. pt. P(2, - 6) and terminal pt. Q(3, 6). 2.) Write the vector in component form. 3.) Sketch the component vector and label it v. 4.) Show .

  9. Vector Operations COMPENENT VECTORS MAKE THE ABOVE SO EASY!!

  10. Yet more new terminology – RESULTANT VECTOR

  11. NORMALIZATION OF A VECTOR The process of finding a unit vector in the same direction as any vector – This proves handy when we want to change the force acting upon something without changing direction. I.O.W – Divide a component vector’s x and y – coords. By the length of the vector. This makes the new length = 1. You are scaling your vector down to an easy-to-use size!

  12. Vectors Written as Linear Combinations A convenient notation often used in applications. YOU ARE EXPECTED TO BE ABLE TO WORK SEEMLESSLY BETWEEN COMPONENT AND LINEAR FORMS OF VECTORS. Represented graphically it becomes clear that i and j are horizontal and vertical components respectively.

  13. Demonstrate the triangle inequality with the following vectors:

  14. We can use trig. And linear combinations to represent vectors involving angles measured counter-clockwise from the positive x-axis. Physics any one? Unit circle: Any vector v with magnitude : Draw out a triangle to see these are our horizontal and vertical components!!!

  15. Represent a component vector with magnitude 8 that makes an angle of 150 degrees with the positive x – axis as: 0.) Linear Combination 1.) Polar Coordinates 2.) Rectangular Coordinates 3.) Component Vector form

  16. Applications to Physics Forces with magnitudes of 500 lbs. and 200 lbs. pull on a box of gold bricks at angles of 30 degrees and –45 degrees with respect to the positive x –axis. Find the direction and magnitude of the resultant force.

  17. Application: Example 7 A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? N E

  18. Application: Example 7 A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? N E u

  19. An application! A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? N v 60o E u

  20. Application: Example 7 A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? N We need to find the magnitude and direction of the resultant vectoru + v. v u+v E u

  21. N The component forms of u and v are: v 70 u+v E 500 u Therefore: and:

  22. N 538.4 6.5o E The new ground speed of the airplane is about 538.4 mph, and its new direction is about 6.5o north of east. p

  23. A real thinker of a problem!! Find a unit vector (a) parallel to and (b) normal to the graph of f(x) at the given point below. Sketch a graph of the vectors and the function.

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