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L 2 – Vectors and Scalars Outline Physical quantities - vectors and scalars

L 2 – Vectors and Scalars Outline Physical quantities - vectors and scalars Addition and subtraction of vector Resultant vector Change in a vector quantity, calculating relative and resolve a vector into components V ector representation in a component form in a coordinate system.

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L 2 – Vectors and Scalars Outline Physical quantities - vectors and scalars

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  1. L 2 – Vectors and Scalars Outline • Physical quantities - vectors and scalars • Addition and subtraction of vector • Resultant vector • Change in a vector quantity, calculating relative andresolve a vector into components • Vector representation in a component form in a coordinate system

  2. Vectors in Physics-Examples Many physical quantities have both magnitude and direction: they are called vectors. • Examples: displacement, velocity, acceleration force, momentum... Other physical quantities have only magnitude: they are called scalars. • Examples: distance, speed, mass, energy...

  3. Displacement and Distance • Displacement is the vector connecting a starting point A and some final point B B A • Distance is the length one would travel from point A to the final point B. Therefore distance is a scalar

  4. Geometrical Representation of Vectors • Arrows on a plane or space • To indicate a vector we use bold letters or an arrow on top of a letter

  5. Properties of Vectors • The opposite of a vector a is vector - a • It has the same length but opposite direction • Two vectors a and b are parallel if one is a positive multiple of the other: a =mb, m>0 Example: if a = 3 b, then a is parallel to b (if a = -2 b then a is anti-parallel to b)

  6. Operations: Adding two Vectors When we add two vectors, we get the resultant vector a + b, with the parallelogram rule:

  7. Operations: Adding more vectors • We can add more vectors by pairing themappropriately

  8. Operations: Vector Subtraction • Special case of vector addition • Add the negative of the subtracted vector • a – b = a + (– b)

  9. Components of a Vector • A component is a part or shadow along a given direction • It is useful to use rectangular components • These are the projections of the vector along the x- and y-axes

  10. Components of a Vector, cont. • The x-component of a vector is the projection along the x-axis • ax = a cosθ • The y-component of a vector is the projection along the y-axis • ay = a sinθ • a is the magnitude of vector a • a2= ax2+ ay2

  11. Example 1 Resolve this vector along the x and y axes to find its components respectively.

  12. Example 2 A vector of 15.0 N at 120º to the x-axis is added to the vector in Example 1. Find the x and y components of the resultant vector. 15.0 N 10.0 N

  13. The Unit Vectors: i, j, k A unit vector has a magnitude of 1 i is the unit vector in the x-direction, j is in the y-direction and k is in the z-direction.

  14. The Unit Vectors, Magnitude • Any vector a can be written as: • a = x i+y j + z k Example 3 : Given the two displacements Show that the magnitude of e is approximately 17 units where:

  15. N - North Direction of Vectors 15 ° east of north, or 75 ° north of east, or bearing of 15 ° ….? 15° 30° E - East W - West 45° 30° …..? 45 ° west of south, or 45 ° south of west, or bearing of 225 ° S - South

  16. Example 4 Find the magnitude and direction of the electric field vector E with components 3i – 4j. Note: This vector could also be written in matrix form:

  17. Relative velocity Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE. A pedestrian (P) is travelling relative to the earth (E) with velocity VPE. • The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by : • VCP= VCE- VPE

  18. Example 5 A boat is heading due north as it crosses a wide river with a velocity of 8.0 km/h relative to water. The river has a uniform velocity of 6.0 km/h due east. Determine the velocity (i.e. speed and direction) of the boat relative to an observer on the riverbank.

  19. The dot (scalar) product • Imagine two vectors a, b at an angle θ • The dot product is defined to be: • a · b = a bcosθ • Useful in finding work of a force F

  20. CHECK LIST • READINGSerway’s Essentials of College Physics pages 41-46 and 53-55.Adams and Allday: 3.3 pages 50-51, 52-53. Summary • Be able to give examples of physical quantities represented by vectors and scalars • Understand how to add and subtract vectors • Know what a resultant vector is • Know how to find the change in a vector quantity, calculate relative andresolve a vector into components • Understand how vectors can be represented in component form in a coordinate system • Be able to do calculations which demonstrate that you have understood the above concepts

  21. Numerical Answers for Examples • Ex 1 – Vx = 8.7N, Vy = 5N • Ex 2 – coordinates of resultant vector are (1.16, 18.0) • Ex 3 – length is 16.9 units, or approximately 17 units • Unknown Directions – Yellow is 60° S of E, or 30 ° E of S, or bearing 150° (90+60) Blue is 30° N of W, or 60 ° W of N, or bearing 300° (270+30) • Ex 4 – Resultant vector E : magnitude 5 units, • Direction 53° S of E, or 37° E of S, or bearing 143° (90 + 53) • Ex 5 – velocity of boat relative to earth: magnitude 10 km/hr • Direction 53° N of E, or 37° E of N, or bearing 37°

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