Vectors Vectors

Physics is the Science of Measurement We begin with the measurement of length: its magnitude and its direction. Length Weight Time

Distance: A Scalar Quantity A scalar quantity: Contains magnitude only and consists of a number and a unit. (20 m, 40 mi/h, 10 gal) A B  Distance is the length of the actual path taken by an object. d = 20 m

Displacement—A Vector Quantity A vector quantity: Contains magnitude AND direction, a number, unit & angle. (12 m, 30 0 ; 8 km/h, N) A B D = 12 m, 20 o Displacement is the straight-line separation of two points in a specified direction.Displacement is the straight-line separation of two points in a specified direction. 

Distance and Displacement Net displacement: 4 m,E 6 m,W D What is the distance traveled? 10 m !! D = 2 m, W Displacement is the x or y coordinate of position. Consider a car that travels 4 m, E then 6 m, W.Displacement is the x or y coordinate of position. Consider a car that travels 4 m, E then 6 m, W. x= +4 x = +4 x= -2 x = -2

Identifying Direction A common way of identifying direction is by reference to East, North, West, and South. (Locate points below.) 40 m, 50 o N of E EW S N 40 m, 60 o N of W 40 m, 60 o W of S 40 m, 60 o S of E Length = 40 m 50 o 60 o

Identifying Direction Write the angles shown below by using references to east, south, west, north. EW S N 45 o EW N 50 o S Click to see the Answers S of E 45 0 W of N

Vectors and Polar Coordinates Polar coordinates (R,  ) are an excellent way to express vectors. Consider the vector 40 m, 50 0 N of E, for example. 0o0o 180 o 270 o 90 o  0o0o 180 o 270 o 90 o R R is the magnitude and  is the direction. 40 m 50 o

Vectors and Polar Coordinates (R,  ) = 40 m, 50 o (R,  ) = 40 m, 120 o (R,  ) = 40 m, 210 o (R,  ) = 40 m, 300 o 50 o 60 o 0o0o 180 o 270 o 90 o 120 o Polar coordinates (R,  ) are given for each of four possible quadrants: 210 o 300 0

Rectangular Coordinates Right, up = (+,+) Left, down = (-,-) (x,y) = (?, ?) x y (+3, +2) (-2, +3) (+4, -3) (-1, -3) Reference is made to x and y axes, with + and - numbers to indicate position in space

Trigonometry Review Application of Trigonometry to VectorsApplication of Trigonometry to Vectors y x R  y = R sin  x = R cos  R 2 = x 2 + y 2 Trigonometry

SOHCAHTOA SOH: sin θ = opp/hypSOH: sin θ = opp/hyp CAH: cos θ = adj/hypCAH: cos θ = adj/hyp TOA: tan θ = opp/adjTOA: tan θ = opp/adj y x R  opposite adjacent hypotenuse

Example 1: Find the height of a building if it casts a shadow 90 m long and the indicated angle is 30 o. 90 m 30 0 The height h is opposite 30 0 and the known adjacent side is 90 m. h h = (90 m) tan 30 o h = 57.7 m

Finding Components of Vectors A component is the effect of a vector along other directions. The x and y components of the vector (R,  are illustrated below. x y R  x = R cos  y = R sin  Finding components: Polar to Rectangular Conversions

Example 2: A person walks 400 m in a direction of 30 o N of E. How far is the displacement east and how far north? x y R  x = ? y = ? 400 m   E N The y-component (N) is OPP: The x-component (E) is ADJ: x = R cos  y = R sin  E N

Example 2 (Cont.): A 400-m walk in a direction of 30 o N of E. How far is the displacement east and how far north? x = R cos  x = (400 m) cos 30 o = +346 m, E x = ? y = ? 400 m   E N Note: x is the side adjacent to angle 30 0 ADJ = HYP x Cos 30 0 The x-component is: R x = +346 m

Example 2 (Cont.): A 400-m walk in a direction of 30 o N of E. How far is the displacement east and how far north? y = R sin  y = (400 m) sin 30 o = m, N x = ? y = ? 400 m   E N OPP = HYP x Sin 30 0 The y-component is: R y = +200 m Note: y is the side opposite to angle 30 0

Example 2 (Cont.): A 400-m walk in a direction of 30 o N of E. How far is the displacement east and how far north? R x = +346 m R y = +200 m 400 m   E N The x- and y- components are each + in the first quadrant Solution: The person is displaced 346 m east and 200 m north of the original position.

Signs for Rectangular Coordinates First Quadrant: R is positive (+) 0 o >  < 90 o x = +; y = + x = R cos  y = R sin  + + 0o0o 90 o R 

Signs for Rectangular Coordinates Second Quadrant: R is positive (+) 90 o >  < 180 o x = - ; y = + x = R cos  y = R sin  + R  180 o 90 o

Signs for Rectangular Coordinates Third Quadrant: R is positive (+) 180 o >  < 270 o x = - y = - x = R cos  y = R sin  - R  180 o 270 o

Signs for Rectangular Coordinates Fourth Quadrant: R is positive (+) 270 o >  < 360 o x = + y = - x = R cos  y = R sin  360 o + R  270 o

Resultant of Perpendicular Vectors Finding resultant of two perpendicular vectors is like changing from rectangular to polar coord. R is always positive;  is from + x axis x y R 

Significant Digits for Angles 40 lb 30 lbR   RyRy RxRx 40 lb 30 lb R  RyRy RxRx  = 36.9 o ; o Since a tenth of a degree can often be significant, sometimes a fourth digit is needed. Rule: Write angles to the nearest tenth of a degree. See the two examples below: