VIEWS: 67 PAGES: 4 CATEGORY: Other POSTED ON: 6/7/2009 Public Domain
Summary Sheet for Vectors in Geometry While a scalar quantity has only a single dimension (length/distance, speed, etc), a vector quantity has two properties: magnitude (the absolute value of its length/ distance) and direction (expressed as an angle measured counterclockwise from standard position on a Cartesian or polar plane or as a navigational bearing measured clockwise from North). Vectors obey the same properties of mathematics as do most other sets of numbers; i.e., commutative, associative, distributive, identity, inverse, addition, subtraction, scalar multiplication and division, etc. A resultant vector is the sum of two or more vectors using either the head to tail method or the parallelogram method. To compose two vectors in a plane written in component form into a resultant vector, we add their corresponding components and employ the Pythagorean Theorem to determine the magnitude of the resultant. We determine the direction of the vector y from standard position by computing the Tan-1 of the components. x x PLEASE NOTE: while the component form of the vector has the pattern , we y y must compute the Tan-1 in order to find the angle of the vector in standard x position. Do not confuse the order of the components. To resolve a vector in polar form into its horizontal (x) and vertical (y) components, we apply the basic rules of trigonometry, where M = Magnitude and is direction in x = M cos() standard position. Thus, vp = M , becomes vc = . y = M sin() Vectors are usually represented in one of three ways: geometrically as rays or arrows with a fixed starting point (endpoint of the ray or “head” of the vector), a specified length proportional to its magnitude, and a direction; algebraically as an ordered pair or ordered triple whose components correspond to the changes in the coordinates of x and y (and z for ordered triples); trigonometrically as a magnitude and direction of a ray in standard position; navigationally as a speed (the scalar quantity of its magnitude) and a bearing/heading (possibly in 3 dimensions). Examples: O OP is a vector starting at Point O extending in the direction of P. P 3 v1 = -4 is a vector written in component form where 3 is the change in the x 12 values, -4 is the change in the y values, and 12 is the change in the z values. v2 =(( 15 newtons, 67° )) is a force vector written in polar coordinate form. v3 =(( 15 knots, bearing 287° )) or v4 = (( warp 4.9 , bearing 287° mark 7 )) are vectors expressed in navigational terms. Writing Vectors: Several notations are in current use for writing vectors. You need to be able to recognize a vector regardless of the form of the notation used, for they all mean "vector." 1. a single boldface letter can be used for 1. a the name of a vector 2. a single letter with a vector arrow above it designates a vector 2. V 3. a pair of change coordinates in angle brackets is the component form of a vector and may be written horizontally 12 with Δx first and Δy second or as a 3. 12 , 35 stacked vector (internationally 35 preferred) with Δx on top and Δy on bottom 4. a listing in angle brackets of the vector 4. 37 , 71 magnitude and its directional angle in standard position measured counter- clockwise from the positive x-axis in 5. 37 , N 19 E that order constitute the polar form of vector notation 5. a listing in angle brackets of the vector magnitude and its bearing measured clockwise from North in that order constitute the navigational form of vector notation x1 x2 The Dot Product of two vectors v1 = and v2 = , written v1 ● v2 , y1 y 2 x1 x2 then is ● = x1 x2 + y1 y2 . y1 y 2 If the dot product of two vectors equals zero, then the two vectors are perpendicular, and are called orthogonal vectors. Two non-zero vectors are parallel, only if one is a uv scalar multiple of the other. cos( ) To compute the angle between two vectors u v (where 0 180°), we can use the formula to the right, where |u| and |v| are the magnitudes of the two vectors. x1 x2 The Cross Product of two vectors v1 = and v2 = , written v1 v2, y1 y 2 represents the vector perpendicular to the plane containing the two vectors and is calculated as the determinant of the components of the two vectors. x1 x2 Hence, v1 v2 = Det = ( x1y2 – y1x2 ) y1 y2 v1 v2 if and only if v1 v2 = 0. Include personal examples here x Often vectors in form are defined in terms of a third variable (such as time). y Such a third variable is called a parameter and usually denoted with the letter t. Then we can define a position vector equation in terms of this parameter as follows: x x x0 y = y t+ y0 x x0 where y is the position of a particle/object t seconds into the journey, y is 0 x the initial position (or starting point), and is the velocity vector. y Of course, we can always rewrite any position vector equation as two parametric equations: x = x ( t ) + x0 and y = y ( t ) + y0 Further, if the vector were originally given in polar form, then we could write the position vector equation as x M cos x0 y = t+ y M sin 0 to produce the two parametric equations x = M cos ( t ) + x0 and y = M sin ( t ) + y0 If we wish to add two vector equations algebraically, where v1& v2 are defined x1 x1 x2 x2 as v1= t + y and v2= t + y then the resultant vector y1 1 y2 2 would keep the initial position of the first vector and add vector coefficients of the two vectors. Thus, x1 x2 x1 v1+ v2 = (t)+ ( t ) + y becomes y1 y2 1 x1+ x2 x1 = (t) + y y1+ y1 1 Practical Vectors with Determinants: If we create a 2 x 2 vector determinant where the vector components are the consecutive sides of a parallelogram in a plane from the same vertex, then the value of the Det x1 x2 determinant is the area of the parallelogram. y1 y2 If we create a 3 x 3 vector determinant where the vector components are the three dimensional vectors from the x1 x2 x3 same vertex, then the value of the determinant is the volume of the parallelepiped. Det y1 y2 y3 z1 z2 z3