Surveyors use accurate measures of magnitudes and directions to create scaled maps of large regions. Vectors
Identifying Direction N 50o 60o E W 60o 60o S A common way of identifying direction is by reference to East, North, West, and South. (Locate points below.) Length = 40 m 40 m, 50o N of E 40 m, 60o N of W 40 m, 60o W of S 40 m, 60o S of E
Identifying Direction N 45o N E W E W 50o S S Write the angles shown below by using references to east, south, west, north. 500 S of E 450 W of N
90o 90o R q 50o 180o 180o 0o 0o 270o 270o Vectors and Polar Coordinates Polar coordinates (R,q) are an excellent way to express vectors. Consider the vector 40 m, 500 N of E, for example. 40 m R is the magnitude and q is the direction.
90o 0o 180o 50o 60o 60o 60o 3000 210o 270o 120o Vectors and Polar Coordinates Polar coordinates (R,q) are given for each of four possible quadrants: = 40 m, 50o = 40 m, 120o = 40 m, 210o = 40 m, 300o
y (-2, +3) (+3, +2) + + x - Right, up = (+,+) Left, down = (-,-) (x,y) = (?, ?) - (+4, -3) (-1, -3) Rectangular Coordinates Reference is made to x and y axes, with + and -numbers to indicate position in space.
R y q x Trigonometry Review Application of Trigonometry to Vectors Trigonometry y = R sin q x = R cos q R2 = x2 + y2
q Finding Components of Vectors A component is the effect of a vector along other directions. The x and y components of the vector are illustrated below. = A cosq = A sin q Finding components: Polar to Rectangular Conversions
Example 2:A person walks 400.0 m in a direction of 30.0oSof W (210o). How far is the displacement westand how far south? 30o 400 m The x-component (W) is adjacent: = -A cosq The y-component (S) is opposite: = -A sinq
Vector Addition Resultant ( ) - sum of two or more vectors. Vector Resolutions: 01. GRAPHICAL SOLUTION – use ruler and protractor to draw and measure the scaled magnitude and angle (direction), respectively. 02. ANALYTICAL SOLUTION - use trigonometry
q Example 11:A bike travels 20 m, E then 40 m at 60o N of W, and finally 30 m at 210o. What is the resultant displacement graphically? Graphically, we use ruler and protractor to draw components, then measure the Resultant R,q C = 30 m B = 40 m 30o R 60o f A = 20 m, E R = (32.6 m, 143.0o) Let 1 cm = 10 m
Cy 30o R Ry 60o f 0 q Ax Rx Bx Cx A Graphical Understanding of the Components and of the Resultant is given below: Note: Rx = Ax + Bx + Cx By B Ry = Ay + By + Cy C A
Resultant of Perpendicular Vectors Finding resultant of two perpendicular vectors is like changing from rectangular to polar coord. R y q x R is always positive; q is from + x axis
First Consider A + B Graphically: B B R B A A Vector Difference For vectors, signs are indicators of direction. Thus, when a vector is subtracted, the sign (direction) must be changed before adding. R = A + B
Now A – B: First change sign (direction) of B, then add the negative vector. B B -B A R’ -B A A Vector Difference For vectors, signs are indicators of direction. Thus, when a vector is subtracted, the sign (direction) must be changed before adding.
Subtraction results in a significant difference both in the magnitude and the direction of the resultant vector. |(A – B)| = |A| - |B| Comparison of addition and subtraction of B B B A R R’ -B B A A Addition and Subtraction R = A + B R’ = A - B
A – B; B - A +A -A +B -B A 2.43 N B 7.74 N Example 13.Given A = 2.4 km, N and B = 7.8 km, N: find A – B and B – A. A - B B - A R R (2.43 N – 7.74 S) (7.74 N – 2.43 S) 5.31 km, S 5.31 km, N