Vectors Scalars a physical quantity described by a

Vectors Scalars: a physical quantity described by a single number Vector: a physical quantity which has a magnitude (size) and direction. • Examples: velocity, acceleration, force, displacement. • A vector quantity is indicated by bold face and/or an arrow. • The magnitude of a vector is the “length” or size (in appropriate units). The magnitude of a vector is always positive. • The negative of a vector is a vector of the same magnitude put opposite direction (i. e. antiparallel) Phys 211 C 1 V p 1

Combining scalars and vectors cannot be added or subtracted. the product of a vector by a scalar is a vector x=ca x = |c| a (note combination of units) if c is positive, x is parallel to a if c is negative, x is antiparallel to a Phys 211 C 1 V p 2

Vector addition most easily visualized in terms of displacements Let X = A + B + C graphical addition: place A and B tip to tail; X is drawn from the tail of the first to the tip of the last A+B=B+A B A X X A B Phys 211 C 1 V p 3

Vector Addition: Graphical Method of R = A + B • Shift B parallel to itself until its tail is at the head of A, retaining its original length and direction. • Draw R (the resultant) from the tail of A to the head of B. A B B + = A = R the order of addition of several vectors does not matter C C D B B C A B A D Phys 211 C 1 V p 4

Vector Subtraction: the negative of a vector points in the opposite direction, but retains its size (magnitude) -B • A- B = A +( -B) R A - B = A + -B = A Phys 211 C 1 V p 5

Resolving a Vector (2 -d) replacing a vector with two or more (mutually perpendicular) vectors => components directions of components determined by coordinates or geometry. A = Ax + Ay Ax = x-component Ay = y-component A Ay q Ax A Ay Be careful in 3 rd , 4 th quadrants when using inverse trig functions to find q. Component directions do not have to be horizontalvertical! q Ax Phys 211 C 1 V p 6

Vector Addition by components R=A+B+C Resolve vectors into components(Ax, Ay etc. ) Add like components Ax + B x + C x = R x Ay + B y + C y = R y The magnitude and direction of the resultant R can be determined from its components. in general R ¹ A + B + C Example 1 -7: Add the three displacements: 72. 4 m, 32. 0° east of north 57. 3 m, 36. 0° south of west 72. 4 m, straight south Phys 211 C 1 V p 7

Unit Vectors a unit vector is a vector with magnitude equal to 1 (unit-less and hence dimensionless) in the Cartesian coordinates: Right Hand Rule for relative directions: thumb, pointer, middle for i, j, k. Express any vector in terms of its components: Phys 211 C 1 V p 8

Products of vectors (how to multiply a vector by a vector) Scalar Product (aka the Dot Product) fis the angle between the vectors A. B = Ax Bx +Ay By +Az Bz = B. A = B cos f A is the portion of B along A times the magnitude of A = A cos f B is the portion of A along B times the magnitude of B B A f B cosf note: the dot product between perpendicular vectors is zero. Phys 211 C 1 V p 9

Example: Determine the scalar product between A = (4. 00 m, 53. 0°) and B = (5. 00 m, 130. 0°) Phys 211 C 1 V p 10

Products of vectors (how to multiply a vector by a vector) Vector Product (aka the Cross Product) 3 -D always! fis the angle between the vectors Right hand rule: A´B = C A – thumb B – pointer C – middle Cartesian Unit vectors C = AB sin f = B sinf A is the portion of B perpendicular A times the magnitude of A = A sinf B is the portion of A perpendicular B times the magnitude of B Phys 211 C 1 V p 11

C = AB sin f = B sin f A is the part of B perpendicular A times A = A sin f B is the part of A perpendicular B times B B sinf B A f Write vectors in terms of components to calculate cross product Phys 211 C 1 V p 12

Example: A is along the x-axis with a magnitude of 6. 00 units, B is in the x-y plane, 30° from the x-axis with a magnitude of 4. 00 units. Calculate the cross product of the two vectors. Phys 211 C 1 V p 13