Vectors Vectors and Scalars Properties of Vectors Components of a - PDF document

Vectors • Vectors and Scalars • Properties of Vectors • Components of a Vector and Unit Vectors • Homework 1

Vectors and Scalars • Vector - quantity that has magnitude and direction – e.g. displacement, velocity, acceleration, force • Scalar - quantity that has only magnitude – e.g. Time, mass, energy 2

Displacement Vector As a particle moves from A to B along the path repre- sented by the dashed curve, its displacement is the vector shown by the arrow from A to B. 3

Adding Vectors When vector B is added to vector A , the resultant R is the vector that runs from the tail of A to the head of B . 4

Commutative Property of Vector Addition • The vector R resulting from the addition of the vectors A and B is the diagonal of a parallelogram of sides A and B . • Vector addition is commutative, that is A + B = B + A . 5

Associative Property of Vector Addition A +( B + C ) = ( A + B )+ C 6

Subtraction of Vectors • To subtract vector B from vector A , simply add the vector - B to vector A . • The vector - B is equal in magnitude and opposite in direction to the vector B . 7

Components of a Vector A vector A lying in the xy plane can be represented by its component vectors A x and A y . A x = A cos θ A y = A sin θ tan θ = A y � A = A 2 x + A 2 y A x 8

Unit Vectors • The unit vectors i , j , and k are directed along the x, y, and z axes, respectively. • The unit vectors i , j , and k form a set of mutually per- pendicular vectors and the magnitude of each unit vector is one – | i |=| j |=| k |=1 9

Vectors in Component Form A vector A lying in the xy plane has component vectors A x i and A y j where A x and A y are the components of A . A =A x i + A y j 10

Example 1 A small plane leaves an airport on an overcast day and later is sighted 215 km away, in a direction making an angle of 22 ◦ east of north. (a) How far east and north is the airplane from the airport when sighted? (b) Using a coordinate system with the y-axis pointing north and the x-axis east, write the position of the airplane in unit vector notation. 11

Example 1 Solution A small plane leaves an airport on an overcast day and later is sighted 215 km away, in a direction making an angle of 22 ◦ east of north. (a) How far east and north is the airplane from the airport when sighted? N y r y r θ r x x θ = 90 ◦ − 22 ◦ = 68 ◦ r x = r cos θ = (215 km ) cos 68 ◦ = 81 km r y = r sin θ = (215 km ) sin 68 ◦ = 199 km (b) Using a coordinate system with the y-axis pointing north and the x-axis east, write the position of the air- plane in unit vector notation. r = r x i + r y j = (81 km ) i + (199 km ) j 12

Vector Addition Using Components R = A + B R x i + R y j = ( A x i + A y j ) + ( B x i + B y j ) R x i + R y j = ( A x + B x ) i + ( A y + B y ) j R x = A x + B x R y = A y + B y 13

Example 2 Find R = A + B + C where A = 4.2 i - 1.6 j , B = -3.6 i + 2.9 j , and C = -3.7 j . 14

Example 2 Solution Find R = A + B + C where A = 4.2 i - 1.6 j , B = -3.6 i + 2.9 j , and C = -3.7 j . R = R x i + R y j R = ( A x + B x + C x ) i + ( A y + B y + C y ) j R = (4 . 2 − 3 . 6 + 0) i + ( − 1 . 6 + 2 . 9 − 3 . 7) j R = 0 . 6 i − 2 . 4 j 15

Homework Set 5 - Due Mon. Sept. 20 • Read Sections 1.8-1.10 • Do Problems 1.35, 1.44, 1.52 & 1.53 16