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CHAPTER 7

CHAPTER 7. APPLICATIONS OF VECTORS. 7.1 VECTORS AS FORCES. Force is a vector because it has a magnitude and direction associated with it. Formula for calculating force is F=ma

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CHAPTER 7

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  1. CHAPTER 7 APPLICATIONS OF VECTORS

  2. 7.1 VECTORS AS FORCES Force is a vector because it has a magnitude and direction associated with it. Formula for calculating force is F=ma Resultant force is the force obtained when all of the forces acting on an object are added together, also known as net force, the resultant force will determine the direction the object moves and the speed at which it will accelerate Cosine law Sine law

  3. 7.1 VECTORS AS FORCES Equilibrant force is the force vector that is opposite to the resultant vector, it is the force that if applied onto the object will prevent any movement. Forces in equilibrium: a group of vectors in equilibrium would equal the 0 vector, there would be no movement of the object Components of forces All vectors can be resolved into horizontal and vertical components using SOH CAH TOA OA can be resolved into OD (x component) and OE (y component) OD =|OA|cosƟ OE = |OE|cosƟ

  4. 7.1 VECTORS AS FORCES Force is a vector because it has a magnitude and direction associated with it. Formula for calculating force is F=ma Resultant force is the force obtained when all of the forces acting on an object are added together, also known as net force, the resultant force will determine the direction the object moves and the speed at which it will accelerate Cosine law Sine law

  5. 7.2 VELOCITY

  6. 7.2 VELOCITY

  7. 7.3 THE DOT PRODUCT: GEOMETRIC VECTORS • dot product of two geometric vectors a and b is a scalar quantity • a • b = |a||b|cosθ where θ is the angle between the two vectors • a • b > 0 when 0° <θ< 90° (acute angle) • a • b < 0 when 90°<θ< 180° (obtuse angle) • a • b = 0 when θ= 90° (right angle/perpendicular) • a • b =b • a (commutative property) • a • (b + c) =a • b + a • c (distributive property) • a • a =|a|² (magnitudes property/vector dotted with itself) • i • i,j • j, and k • k all/each equal 1 • (ka) • b = a • kb + k(a • b) • for collinear vectors, the dot product is a • b = +|a||b| depending on if θ is either cos0° = 1 or cos180° = -1 a • b cosθ = -------- |a||b|

  8. 7.4 THE DOT PRODUCT: ALGEBRAIC VECTORS • used when we are given COMPONENTS • for R², if a = (a1,a2) and b = (b1,b2) then a • b = a1a2 + b1b2 • for R³, if a = (a1, a2, a3) and b = (b1, b2, b3) then a • b = a1b1 + a2b2 + a3b3 • same properties for geometric vectors hold true here as well

  9. 7.5 SCALAR AND VECTOR PROJECTIONS

  10. 7.5 SCALAR AND VECTOR PROJECTIONS

  11. 7.5 SCALAR AND VECTOR PROJECTIONS

  12. 7.5 SCALAR AND VECTOR PROJECTIONS

  13. 7.6 THE CROSS PRODUCT Cross product for Geometric vectors is equivalent to the area of the parallelogram created by those 2 vectors a X b =|a||b|sinƟ To find the direction of this vector use the right hand rule place your hand on the first vector and curl it along the angle to the second vector, whichever way your thumb is pointing is the direction of the vector Cross product for Algebraic vectors a= (a1,a2,a3) and b = (b1,b2,b3)

  14. 7.6 THE CROSS PRODUCT

  15. 7.7 APPLICATIONS OF THE DOT AND CROSS PRODUCTS WORK TORQUE • dot product (scalar quantity) • product of component of force exerted on object and its displacement • negative answer = displacement opposite to direction of force • cross product (vector quantity) • resulting turning effect due to force directed at an angle with radius measured between exerted force and point of rotation • we only want the MAGNITUDE We can use algebraic model here as well 

  16. REVIEW GAME!

  17. Determine the resultant and the equilibrant for

  18. Are the following vectors in equilibrium?

  19. Determine the resultant and the equilibrant for

  20. A couple is travelling on a bus 50km/h due south. They are travelling parallel to a car, which appears to the couple to be travelling 80km/h due north. What is the actual speed of the car?

  21. An airplane has an airspeed of 300 km/h [N 30° W]. A wind is blowing from the south at 60 km/h. Find the plane’s resultant velocity.

  22. Feist leaves a dock, paddling her canoe at 5 m/s. She heads downstream at an angle of 30 degrees to the current, which is flowing at a rate of 12 m/s. a) How far does he travel downstream in 10 seconds? b) How long does it take to cross to river if it is 180m wide?

  23. Calculate the dot product of these two vectors, given an angle of θ between them: |r| = 7, |s| = 9, θ= 35°

  24. Prove the following: (8a - 3b) • (8a + 3b) = 64|a|² - 9|b|²

  25. Simplify the following expression: 4m • (2m – 5n) + (n + 6m) • (m – 3n)

  26. Calculate the dot product of these two vectors, and then state whether the angle between them is acute, obtuse, or 90° v = ( -6, 7, -8) w = (9, -10, 11)

  27. If a = (d, 6, 13) and b = (-15, 9, d) are perpendicular, what is the value of d?

  28. Gavin and Marissa are arguing about a parallelogram that has diagonals e = (-12, -12, 0) and f = (4, -4, 8). Gavin says the parallelogram is just a rhombus, but Marissa thinks it might also be a square. Who is more correct about the shape of the parallelogram?

  29. What are the scalar and vector projections of x onto y when x = (2,5) and y = (-5, 12)?

  30. In |ab|, does the magnitude of b affect the scalar projection of a onto b?

  31. Calculate the cross product of these two vectors: and

  32. Solve (a x b) + (a x c) where

  33. Solve (a x b) + (a x c) · (c x b) where

  34. Calculate the amount of work done when a block is pushed at an angle 14 degrees above the horizontal for 50 metres with a force of 12 N.

  35. Determine the area of the parallelogram formed by the vectors a = (1,1,0) and b = (1,0,1).

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