Chapter 3 Vectors

Vectors and Scalars A Scalar is a physical quantity with magnitude (and units). Examples: Temperature, Pressure, Distance, Speed A Vector is a physical quantity with magnitude and direction: Displacement: Washington D.C. is ~ 180 miles N of Newport News Wind Velocity: 20mi/hr towards SW

Components of a vector, as an alternate to magnitude and direction 100 miles 30 degrees north of east, is equivalent to 86.6 miles east followed by 50 miles north N 100mi 50mi 86.6mi E

Labels for Components of a Vector To free ourselves from the points of the compass, we will use x & y instead of E & N Vector , magnitude Components (as ordered pair)

Trigonometry and Vector Components Trigonometry is not a pre-requisite for this course. Today you will learn ½ of trigonometry, and all that you need for this course. In this discussion, we always define the direction of a vector in terms of an angle counter-clockwise from the + x-axis. Negative angles are measured clockwise.

Trigonometry and Circles The point P1=(x1,y1) lies on a circle of radius r. The line from the origin to P1 makes an angle q1 w.r.t. the x-axis. The trigonometric functions sine and cosine are defined by the x- and y-components of P1: x1 y1 r P1 q

45-45-90 triangle By symmetry, Pythagorean Theorem: cos(45º) = x1 /r x1 = y1 Pythagorean Theorem: x12 + y12 = r2 2· x12 = r2 x1 = r/2 cos(45º) = x1 /r cos(45º) = 0.7071 sin(45º) = 1/ 2

30-60-90 Triangle

Vector Addition:Graphical (use bold face for vector symbol) A, B, and C are three displacement vectors. Any point can be the origin for a displacement The vector B = 3 paces to E. Notice that B has been translated from the origin until the tail of B is at the head of A. This is the “head-to-tail” method of vector addition. Vector addition is commutative, just like ordinary addition: D = A+B+C = C+B+A

Vector Addition, Components When we add two vectors, the components add separately: Cx = Ax + Bx Cy = Ay + By

Velocity Vectors Each fish in a school has its own velocity vector. If the fish are swimming in unison, the velocity vectors are all (nearly) identical We draw each vector at the position of the fish.

Scalar Multiplication Multiplying a vector by a scalar Multiplying a vector by a positive scalar quantity simply re-scales the length (and maybe units) of the vector, without changing direction. Multiplying a vector by a negative number reverses the direction of the vector. y x

Vector Subtraction Subtraction is just addition of the additive inverse y x

Average Velocity Vector Net displacement (vector) multiplied by reciprocal of elapsed time (scalar) r1 r2

Example 1 A whale comes to the surface to breathe, and then dives at an angle 20.0° below the horizontal. Answer the following questions if the whale continues in a straight line for 140 m. (a) How deep is it? (b) How far has it traveled horizontally?

Example 2 Consider the vectors in Figure 3-36, in which the magnitudes of A, B, C, and D are respectively given by 15 m, 20 m, 10 m, and 15 m. Express the sum, A + C + D, in unit vector notation.

Relative Motion

Relative Motion

Relative Motion Example As an airplane taxies on the runway with a speed of 15.4 m/s, a flight attendant walks toward the tail of the plane with a speed of 1.30 m/s. What is the flight attendant's speed relative to the ground?