Vectors Quantities with direction

Vectors and Scalars Vector: quantity needing a direction to fully specify (direction + magnitude) Scalar: directionless quantity

Represent as Arrows direction: obvious magnitude: length location is irrelevant these are identical

Represent as Components Components: magnitudes in x and y (maybe z) directions (x = right, y = up) B A A = (4, 3) B = (0, –2) x y

Represent as Polar Coordinates x  r

Unit Vectors Essentially direction-only vectors Directions along axes Magnitude is unitless 1 Basis vectors in 3-space unit vector k : in positive z-direction unit vector i : in positive x-direction unit vector j : in positive y-direction

Represent as Basis Combinations C B A A = 4i + 3j B = –2j C = A + B = 4i + (3–2)j = 4i + j

Cartesian to Polar Magnitude r of a vector: use theorem of Pythagoras r 2 = x 2 + y 2 + z 2 r 2 = r = = = = r

Cartesian to Polar Direction  of a vector: use tan (  ) = y/x A tan  = 3/4 A = (4, 3)  = arctan(3/4)  = 36.87° 

Polar to Cartesian Components x and y of a vector: use sine and cosine r y = r sin  x = r cos 

Vector Operations addition and subtraction

Add Vectors Graphically A C B A + B = C Head-to-tail A B

How to Add Vectors Graphicaly Head-to-Tail Place following vector’s tail at preceding vector’s head Resultant starts where the first vector starts and ends where the last vector ends

Add as Components Add components for each direction separately C B A A = (4, 3) B = (0, –2) C = A + B = (4+0, 3–2) = (4, 1)

Subtract Vectors A B Add the negative of the vector being subtracted. –B–B A – B = A + (–B) = D D –B–B A D + B = A

Scalar Multiplication Product of (scalar)(vector) is a vector The scalar multiplies the magnitude of the vector; direction does not change Direction reverses if scalar is negative A 2 A 1/2 A –2 A

Dot Product of Vectors a·b = ab cos  a b a b  Commutative

Dot Product Geometrically Product of the projection of one vector onto the other “Overlap” b cos  a cos  a b  ab cos 

Dot Product by Components A·B = (A x B x + A y B y + A z B z )

Cross Product of Vectors Operation symbol  Another way to multiply two vectors Product is a vector! Direction of A  B is perpendicular to both A and B

Cross Product by Components Determinant method A  B = AxAx AyAy AzAz BxBx ByBy BzBz ijk

Cross Product Magnitude  A  B  = AB sin  A B A B  Maximum for  = 90° Zero for  = 0°, 180°

Magnitude Geometrically A B A B   A  B  = area of parallelogram

Cross Product Direction Curl right-hand fingers in direction of  Right-hand thumb points in direction of cross- product Not commutative A B A B  AB = –(BA)AB = –(BA)