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- 1. SCALAR AND VECTOR QUANTITIES RAPHAEL V. PEREZ, CpE
- 2. SCALAR AND VECTOR QUANTITIES Define Scalar and Vector
- 3. SCALAR QUANTITY has only magnitude. (Only the measure / quantity) VECTOR QUANTITY has both magnitude and direction.
- 4. SCALAR VECTOR distance displacement work power acceleration volume pressure velocity speed weight mass force resistance
- 5. SCALAR AND VECTOR QUANTITIES WHAT IS RESULTANT VECTOR?
- 6. RESULTANT VECTOR is the is the vector that 'results' from adding two or more vectors together. -1 -0.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 0.5 1 1.5 2 2.5 3 3.5 4 y
- 7. The goal of this topic is to find the MAGNITUDE OF THE RESULTANT VECTOR (R), and the VECTOR ANGLE (θ)
- 8. THERE ARE THREE TECHNIQUES TO FIND THE RESULTANT VECTOR AND THE VECTOR ANGLE: 1. GRAPHICAL METHOD – you need the technical tools like sharp pencil, ruler, protractor and the paper (graphing or bond) to show the vectors graphically. The output is the connection of vectors is like a polygon.
- 9. R ɵ
- 10. SAMPLE PROBLEM 1. A man walks at40 meters East and 30 meters North. Find the magnitude of resultant displacement and its vector angle. Use Graphical Method.
- 11. Solution: Write the given facts Given: A = 40 meters East B = 30 meters North R = ? θ = ?
- 12. graph the vectors from the origin (head to tail) head (arrowhead) VECTOR tail
- 13. graph the vectors from the origin (head to tail) NOTE: 1 GRID = 10 METERS B=30METERS, NORTH A = 40 METERS, EAST USE RULER TO MEASURE AND TO DRAW A LINE θ = 37 N of E
- 14. SAMPLE PROBLEM 2. Given: A = 5 km ,East B = 6 km, NE C = 7 km, 30˚ N of W R = ? θ = ?
- 15. graph the vectors from the origin (head to tail) NOTE: 1 GRID = 10 km POSSIBLE GRAPH 2. Given: A = 5 km ,East B = 6 km, NE C = 7 km, 30˚ N of W R = ? θ = ?
- 16. ASSIGNMENT Use graphical Method to find the magnitude of the resultant displacement and the vector angle 1. Given: A= 13cm, 30 N of E B= 20 cm, North R = ? θ = ? 2. Given: M= 5.7 cm, NW N= 2.5 cm, SE O= 1.3 cm, NE R = ? θ = ?
- 17. 2. The Pythagorean Theorem The Pythagorean theorem is a useful method for determining the result of adding two (and only two) vectors that make a right angle to each other. The method is not applicable for adding more than two vectors or for adding vectors that are not at 90- degrees to each other. The Pythagorean theorem is a mathematical equation that relates the length of the sides of a right triangle to the length of the hypotenuse of a right triangle.
- 18. SAMPLE PROBLEM 1. A man walks at 40 meters East and 30 meters North. Find the magnitude of resultant displacement and its vector angle. Use Pythagorean Theorem.
- 19. B=30METERS,NORTH A = 40 METERS, EAST sketch your problem θ
- 20. • 1. ____ is an example of a scalar quantity a) velocity b) force c) volume d) acceleration • 2. ___ is an example of a vector quantity a) mass b) force c) volume d) density
- 21. • 3. A scalar quantity: a) always has mass b) is a quantity that is completely specified by its magnitude c) shows direction d) does not have units • 4. A vector quantity a) can be a dimensionless quantity b) specifies only magnitude c) specifies only direction d) specifies both a magnitude and a direction
- 22. • 5. A boy pushes against the wall with 50 pounds of force. The wall does not move. The resultant force is: a) -50 pounds b) 100 pounds c) 0 pounds d) -75 pounds • 6. A man walks 3 miles north then turns right and walks 4 miles east. The resultant displacement is: a) 1 mile SW b) 7 miles NE c) 5 miles NE d) 5 miles E
- 23. • 7. A man walks at 10 m East, but he returns back at 10 m at west. The resultant displacement is: a) 0 km b) 20 km c) 10 km d) -10 km • 8. The difference between speed and velocity is: a) speed has no units b) speed shows only magnitude, while velocity represents both magnitude (strength) and direction c) they use different units to represent their magnitude d) velocity has a higher magnitude
- 24. • 7. A plane flying 500 MI/hr due north has a tail wind of 45 MI/hr the resultant velocity is: a) 545 miles/hour due south. b) 455 miles/hour north. c) 545 miles/hr due north. d) 455 MI/hr due south • 8. The difference between speed and velocity is: a) speed has no units b) speed shows only magnitude, while velocity represents both magnitude (strength) and direction c) they use different units to represent their magnitude d) velocity has a higher magnitude
- 25. • 9. The resultant magnitude of two vectors a) Is always positive b) Can never be zero c) Can never be negative d) Is usually zero • 10. Which of the following is not true. a) velocity can be negative b) velocity is a vector b) speed is a scalar d) speed can be negative
- 26. 3. ANALYTICAL (COMPONENT) METHOD Each vector has two components : the x-component and the y-component If the vectors are in secondary directions : (NW, NE, SW or SE directions) Ax = A cos θx Ay = A sin θx where: A = the given vector value θx = the given angle from x -axis Ax = the x – component of vector A Ay = y – component of vector A
- 27. Component formula for x and y: Ax = A cos θx Ay = A sin θx Sum of x and y Components:
- 28. Consider the sign conventions for the Sum of x and y Components Quadrant of Magnitude + + I - + II - - III + - IV 0 Y-axis (North or South) 0 X- axis (West or East)
- 29. About the vector angle: Recall it from Trigonometry: if θx is positive: rotation of magnitude is counterclockwise from x-axis if θx is negative: rotation of magnitude is clockwise from x-axis
- 30. Recall the SAMPLE PROBLEM Given: A = 5 km ,East B = 6 km, NE C = 7 km, 30˚ N of W R = ? θ = ?
- 31. Solution: Draw a sketch A = 5 km ,East B = 6 km, NE C = 7 km, 30˚ N of W
- 32. Solution: Draw a table Vector and measure X-component Y-component A +5 0 B +6 cos 45 +6 sin 45 C -7 cos30 +7 sin 30 Total (use scientific calculator in degrees mode) Note: for the sum of components: round off the answers into 5 decimal places. Therefore, the direction of the magnitude of resultant vector is in QUADRANT I
- 33. Solution: Compute for the magnitude and vector angle WHICH IS NEAR IN OUR PREVIOUS DRAWING IN GRAPHICAL METHOD
- 34. Actual happen on vectors (not needed to graph) NOTE: 1 GRID = 10 km POSSIBLE GRAPH A = 5 km ,East B = 6 km, NE C = 7 km, 30˚ N of W R = ? θ = ?
- 35. graph the vectors from the origin (head to tail)
- 36. θ = 67.67 (ƩRx, ƩRy) = (3.18046, 7.74264) Quadrant I
- 37. FINAL ANSWER
- 38. 4 N Vectors X- component Y- component A B C Total A = B = C =
- 39. A = B = C = D = 7 N Vectors X-component Y-component A B C D Total

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