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PHY1012F VECTORS

PHY1012F VECTORS. Gregor Leigh gregor.leigh@uct.ac.za. VECTORS. Resolve vectors into components and reassemble components into a single vector with magnitude and direction. Make use of unit vectors for specifying direction.

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PHY1012F VECTORS

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  1. PHY1012FVECTORS Gregor Leighgregor.leigh@uct.ac.za

  2. VECTORS • Resolve vectors into components and reassemble components into a single vector with magnitude and direction. • Make use of unit vectors for specifying direction. • Manipulate vectors (add, subtract, multiply by a scalar) both graphically (geometrically) and algebraically. Learning outcomes:At the end of this chapter you should be able to…

  3. VECTORS • Using only positive and negative signs to denote the direction of vector quantities is possible only when working in a single dimension (i.e. rectilinearly). • In order to deal with direction when describing motion in 2-d (and later, 3-d) we manipulate vectors using either graphical (geometrical) techniques, or the algebraic addition of vector components.

  4. SCALARS and VECTORS • Scalar – A physical quantity with magnitude (size) but no associated direction.E.g. temperature, energy, mass. Vector – A physical quantity which has both magnitude AND direction.E.g. displacement, velocity, force.

  5. Algebraically, we shall distinguish a vector from a scalar by using an arrow over the letter, . VECTOR REPRESENTATION and NOTATION • Graphically, a vector is represented by a ray.The length of the ray represents the magnitude, while the arrow indicates the direction. The important information is in the direction and length of the ray – we can shift it around if we do not change these. Note: r is a scalar quantity representing the magnitude of vector , and can never be negative.

  6. GRAPHICAL VECTOR ADDITION • A helicopter flies 15 km on a bearing of 10°, then 20 km on a bearing of 250°. Determine its net displacement. N 20 km 60° 10° 15 km   = 74°

  7. MULTIPLYING A VECTOR BY A SCALAR • Multiplying a vector by a positive scalar gives another vector with a different magnitude but the same direction: Notes: • B = cA. (c is the factor by which the magnitude ofis changed.) • lies in the same direction as . • (Distributive law). • If c is zero, the product is the directionless zero vector, or null vector.

  8. y • x • z VECTOR COMPONENTS • Manipulating vectors geometrically is tedious. • Using a (rectangular) coordinate system, we can use components to manipulate vectors algebraically. • We shall use Cartesian coordinates, a right-handed system of axes: (The (entire) system can be rotated – any which way – to suit the situation.)

  9. VECTOR COMPONENTS • Adding two vectors (graphically joining them head-to-tail) produces a resultant (drawn from the tail of the first to the head of the last)…

  10. VECTOR COMPONENTS • Adding two vectors (graphically joining them head-to-tail) produces a resultant (drawn from the tail of the first to the head of the last)… “Running the movie backwards” resolves a single vector into two (or more!) components.

  11. VECTOR COMPONENTS • Adding two vectors (graphically joining them head-to-tail) produces a resultant (drawn from the tail of the first to the head of the last)… “Running the movie backwards” resolves a single vector into two (or more!) components.

  12. VECTOR COMPONENTS • Adding two vectors (graphically joining them head-to-tail) produces a resultant (drawn from the tail of the first to the head of the last)… “Running the movie backwards” resolves a single vector into two (or more!) components. ??! Even if the number of components is restricted, there is still an infinite number of pairs into which a particular vector may be decomposed. Unless…

  13. VECTOR COMPONENTS • …by introducing axes, we specify the directions of the components. • y is now constrained to resolve into and , at right angles to each other. Note that, provided that we adhere to the right-handed Cartesian convention, the axes may be orientated in any way which suits a given situation. • x

  14. y • x VECTOR COMPONENTS • Resolution can also be seen as a projection of onto each of the axes to produce vector components and . Ax, the scalar component of (or, as before, simply its component) along the x-axis … • has the same magnitude as . • is positive if it points right; negative if it points left. • remains unchanged by a translation of the axes (but is changed by a rotation).

  15. y (m) 8 6 4 2 • x (m) 0 0 2 4 6 8 VECTOR COMPONENTS • The components of are… Ax = +6 m Ay = +3 m

  16. VECTOR COMPONENTS • y (m) • The components of are… 8 6 Ax = +6 m Ay = +3 m 4 2 • x (m) -8 -6 -4 -2 0

  17. VECTOR COMPONENTS • y (m) • The components of are… -8 -6 -4 -2 • x (m) -2 Ax = +6 m Ay = +3 m -4 -6 -8

  18. VECTOR COMPONENTS • y (m) • The components of are… 4 2 Ax = –2 m Ay = +4 m • x (m) -4 -2 2 -2

  19. VECTOR COMPONENTS • y (m) • The components of are… 4 2 Ax = –6 m Ay = –5 m • x (m) -8 -4 4 -2

  20. VECTOR COMPONENTS The components of are… PHY1012F • y (m) 4 • x (m) 2 Ax = –6 m Ay = –5 m –8 m –3 m 4 -4 -8 -4 20

  21. PHY1010W VECTOR COMPONENTS • y (m) • The components of are… • x (m) Ax = +Acos m Ay = –Asin m • 

  22. PHY1010W VECTOR COMPONENTS • y (m) • The components of are… • x (m) Ax = –Asin m Ay = –Acos m •  Note that we can (re)combine components into a single vector, i.e. (re)write it in polar notation, by calculating its magnitude and direction using Pythagoras and trigonometry: (On this slide!)

  23. y 1 • x 1 UNIT VECTORS • Components are most useful when used with unit vector notation. • A unit vector is a vector with a magnitude of exactly 1 pointing in a particular direction: • A unit vector is pure direction – it has no units! • Vector can now be resolved and written as:

  24. UNIT VECTORS • y (m/s) Given a 12 m/s velocity vector which makes an angle of 60° with the negative x-axis, write the vector in terms of components and unit vectors. vy = +vsin60° v = 12 m/s • 60° vx = –(12 m/s)cos60° vx= –6.00 m/s • x (m/s) vx = –vcos60° vy = +(12 m/s)sin60° vy= +10.4 m/s Hence:

  25. ALGEBRAIC ADDITION OF VECTORS • Suppose and Thus Dx= Ax + Bx + Cx and Dy= Ay + By + Cy In other words, we can add vectors by adding their components, axis by axis, to determine a single resultant component in each direction. These resultants can then be combined, or simply presented in unit vector notation.

  26. y • x ALGEBRAIC ADDITION OF VECTORS • The process of vector addition by the addition of components can visualised as follows: By Dy = Ay + By Ay = + Ax Bx Dx = Ax + Bx

  27. y • x ALGEBRAIC ADDITION OF VECTORS • While it is often quite acceptable to present as Dy = Ay + By its polar form is easily reconstituted from Dx and Dy using and •  Dx = Ax + Bx

  28. GRAPHICAL VECTOR ADDITION • A helicopter flies 15 km on a bearing of 10°, then 20 km on a bearing of 250°. Determine its net displacement. N 20 km 60° 10° 15 km   = 74°

  29. ALGEBRAIC ADDITION OF VECTORS • A helicopter flies 15 km on a bearing of 10°, then 20 km on a bearing of 250°. Determine its net displacement. • y N 20 km +15 sin10° +2.60 +15 cos10° +14.77 10° –18.79 –20 cos20° –20 sin20° –6.84 15 km –16.2 +7.93  • x 20°

  30. N • y • x ALGEBRAIC ADDITION OF VECTORS • A spelunker is surveying a cave. He follows a passage 100 m straight east, then 50 m in a direction 30° west of north, then 150 m at 45° west of south. After a fourth unmeasured displacement he finds himself back where he started. Determine the magnitude and direction of his fourth displacement. 30° 50 m 100 m 150 m 45°

  31. ALGEBRAIC ADDITION OF VECTORS • y N Rx = 100 – 25 –106 + Dx = 0  Dx = 31 30° Ry = 0 +43.3 –106 + Dy = 0  Dy = 62.7 50 m • x 100 m 150 m 45° 100 0 –25 43.3 –106 –106 63.7 69.9 ? ? ? 31 62.7 ?

  32. VECTORS • Resolve vectors into components and reassemble components into a single vector with magnitude and direction. • Make use of unit vectors for specifying direction. • Manipulate vectors (add, subtract, multiply by a scalar) both graphically (geometrically) and algebraically. Learning outcomes:At the end of this chapter you should be able to…

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