Chapter 1 – Math Review

Vectors Surveyors use accurate measures of magnitudes and directions to create scaled maps of large regions.

Objectives: After completing this module, you should be able to: Demonstrate that you meet mathematics expectations: unit analysis, algebra, scientific notation, and right-triangle trigonometry. Define and give examples of scalar and vector quantities. Determine the components of a given vector. Find the resultant of two or more vectors.

Expectations You must be able convert units of measure for physical quantities. Convert 40 m/s into kilometers per hour. 40--- x ---------- x -------- = 144 km/h m s 1 km 1000 m 3600 s 1 h

Expectations (Continued) You must be able to work in scientific notation. Evaluate the following: (6.67 x 10-11)(4 x 10-3)(2) (8.77 x 10-3)2 F = -------- = ------------ Gmm’ r2 F = 6.94 x 10-9 N = 6.94 nN

Expectations (Continued) You must be familiar with SI prefixes The meter (m) 1 m = 1 x 100 m 1 Gm = 1 x 109 m 1 nm = 1 x 10-9 m 1 Mm = 1 x 106 m 1 mm = 1 x 10-6 m 1 km = 1 x 103 m 1 mm = 1 x 10-3 m

Expectations (Continued) You must have mastered right-triangle trigonometry. y x R q y = R sin q x = R cos q R2 = x2 + y2

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

Some Physics Quantities Vector - quantity with both magnitude (size) and direction Scalar - quantity with magnitude only Vectors: Displacement Velocity Acceleration Momentum Force Scalars: Distance Speed Time Mass Energy

Mass vs. Weight Mass Scalar (no direction) Measures the amount of matter in an object Weight Vector (points toward center of Earth) Force of gravity on an object On the moon, your mass would be the same, but the magnitude of your weight would be less.

Vectors are represented with arrows The length of the arrow represents the magnitude (how far, how fast, how strong, etc, depending on the type of vector). The arrow points in the directions of the force, motion, displacement, etc. It is often specified by an angle. 5 m/s 42°

Units are not the same as quantities! Quantity . . . Unit (symbol) Displacement & Distance . . . meter (m) Time . . . second (s) Velocity & Speed . . . (m/s) Acceleration . . . (m/s2) Mass . . . kilogram (kg) Momentum . . . (kg · m/s) Force . . .Newton (N) Energy . . . Joule (J)

SI Prefixes Little Guys Big Guys

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

Displacement—A Vector Quantity Displacement is the straight-line separation of two points in a specified direction. A vector quantity: Contains magnitude AND direction, a number, unit & angle. (12 m, 300; 8 km/h, N) A B D = 12 m, 20o q

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

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

Identifying Direction Write the angles shown below by using references to east, south, west, north. E W N 50o S E W S N 45o 500 S of E Click to see the Answers . . . 450 W of N

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. 0o 180o 270o 90o q 0o 180o 270o 90o R 40 m 50o R is the magnitude and q is the direction.

Vectors and Polar Coordinates Polar coordinates (R,q) are given for each of four possible quadrants: 0o 180o 270o 90o (R,q) = 40 m, 50o 120o 60o 210o 50o (R,q) = 40 m, 120o 3000 60o 60o (R,q) = 40 m, 210o (R,q) = 40 m, 300o

Example 1: Find the height of a building if it casts a shadow 90 m long and the indicated angle is 30o. 90 m 300 The height h is opposite 300 and the known adjacent side is 90 m. h h = (90 m) tan 30o 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,q) are illustrated below. x = R cos q y = R sin q x y R q Finding components: Polar to Rectangular Conversions

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

Note: x is the side adjacent to angle 300 Example 2 (Cont.): A 400-m walk in a direction of 30o N of E. How far is the displacement east and how far north? x = ? y = ? 400 m 30o E N Note: x is the side adjacent to angle 300 ADJ = HYP x Cos 300 x = R cos q The x-component is: Rx = +346 m x = (400 m) cos 30o = +346 m, E

Note: y is the side opposite to angle 300 Example 2 (Cont.): A 400-m walk in a direction of 30o N of E. How far is the displacement east and how far north? x = ? y = ? 400 m 30o E N Note: y is the side opposite to angle 300 OPP = HYP x Sin 300 y = R sin q The y-component is: Ry = +200 m y = (400 m) sin 30o = + 200 m, N

The x- and y- components are each + in the first quadrant Example 2 (Cont.): A 400-m walk in a direction of 30o N of E. How far is the displacement east and how far north? Rx = +346 m Ry = +200 m 400 m 30o 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 (+) 0o > q < 90o x = +; y = + R + q 0o + x = R cos q y = R sin q

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

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

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

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