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6.1 Vectors in the Plane

6.1 Vectors in the Plane. Some quantities can be represented by numbers only: Temperature, height, distance, volume, speed, area. These numbers indicate the size or magnitude . Other quantities have both size (magnitude) and direction: velocity, acceleration, force. Vector.

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6.1 Vectors in the Plane

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  1. 6.1 Vectors in the Plane

  2. Some quantities can be represented by numbers only: Temperature, height, distance, volume, speed, area. These numbers indicate the size or magnitude. • Other quantities have both size (magnitude) and direction: velocity, acceleration, force.

  3. Vector We represent magnitude and direction with an arrow (directed line segment) whose tail is at the origin and an ordered pair determines the head. This is the position vector of and is denoted by or

  4. is known as the component form of vector • Vectors do not have to originate from the origin. As long as two vectors have the same length and same direction they are equivalent.

  5. and have the same length and direction even though it is in a different location (position). and are equivalent and B A D C

  6. Terminal – initial Or Head – tail

  7. Given the points and B, the vector with representation is Ex: Find the vector represented by the directed line segment with initial point and terminal point

  8. Magnitude • If is represented by the arrow from to then If then

  9. Examples • Find the magnitude of where and • Find the component form of

  10. The Sum of Two Vectors If a particle moves from A to B then changes direction and moves to C, the resulting displacement is from A to C. B A is the resultant C

  11. Definition of Vector Addition If and are vectors positioned so the initial point of is at the terminal point of (tip to tail) then the sum of is the vector from the initial point of to the terminal point of . The resultant (sum) is from the tail of the first to the head of the last.

  12. Definition of scalar multiplication If c is a scalar (magnitude only) and is a vector, then the scalar multiplication is the vector whose length is times the length of and whose direction is the same as if and is opposite to if .

  13. Math with Components

  14. Examples • Find • Find the magnitude of

  15. Vector Operations Find Find How would you draw ?

  16. Given vectors and Show:

  17. Example If and Find:

  18. Unit Vectors A unit vector is a vector of length 1 in the direction of the original vector Find the unit vector of

  19. Day 2 – Vectors

  20. Component form to linear combination -3 is the horizontal component 2 is the vertical component 2 -3

  21. How would we find magnitude and direction?

  22. Components y component ᶿ x component

  23. Example • Find the components of with direction angle of and magnitude of 6 • Find the magnitude and direction of

  24. Velocity vs Speed • Velocity is a vector. It has magnitude and direction. Speed is the magnitude of the velocity – no direction! It is not a vector. It is a scalar only.

  25. Example • A DC – 10 jet aircraft is flying on a bearing of at 500 mph. Find the component form of the velocity of the airplane.

  26. Example

  27. Ex: A 100 lb weight hangs from two wires. Find the tensions T1 and T2 in both wires and their magnitudes. 50° 32° T1 T2 100 lb

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