Phys211C1V p1 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)

Phys211C1V p2 Combining scalars and vectors scalars and vectors cannot be added or subtracted. the product of a vector by a scalar is a vector x = c a 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

Phys211C1V p3 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 A B X A B X

Phys211C1V p4 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 += A B = R the order of addition of several vectors does not matter A C B D A B C DD B A C

Phys211C1V p5 Vector Subtraction: the negative of a vector points in the opposite direction, but retains its size (magnitude) A  B = A +(  ) A B  = A BB  R = A BB

Phys211C1V p6 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 AyAy AxAx A = A x + A y A x = x-component A y = y-component  A AyAy AxAx  Be careful in 3 rd, 4 th quadrants when using inverse trig functions to find . Component directions do not have to be horizontal- vertical!

Phys211C1V p7 Vector Addition by components R = A + B + C Resolve vectors into components(A x, A y etc. ) Add like components A x + B x + C x = R x A y + B y + C y = R y The magnitude and direction of the resultant R can be determined from its components. in general R  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

Phys211C1V p8 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:

Phys211C1V p9 Products of vectors (how to multiply a vector by a vector) Scalar Product (aka the Dot Product)  is the angle between the vectors A. B = A x B x +A y B y +A z B z = B. A = B cos  A is the portion of B along A times the magnitude of A = A cos  B is the portion of A along B times the magnitude of B B A  B cos  note: the dot product between perpendicular vectors is zero.

Phys211C1V p10 Example: Determine the scalar product between A = (4.00m, 53.0°) and B = (5.00m, 130.0°)

Phys211C1V p11 Products of vectors (how to multiply a vector by a vector) Vector Product (aka the Cross Product) 3-D always!  is the angle between the vectors Right hand rule: A  B = C A – thumb B – pointer C – middle Cartesian Unit vectors C = AB sin  = B sin  A is the portion of B perpendicular A times the magnitude of A = A sin  B is the portion of A perpendicular B times the magnitude of B

Phys211C1V p12 Write vectors in terms of components to calculate cross product C = AB sin  = B sin  A is the part of B perpendicular A times A = A sin  B is the part of A perpendicular B times B B A  B sin 

Phys211C1V p13 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.