Dave's Math Tables: Vectors |
(Math | OddsEnds | Vectors) |
Notation: The lower case letters a-h, l-z denote scalars.Uppercase bold A-Z denote vectors. Lowercase boldi, j, k denote unit vectors. <a, b>denotes a vector with components a and b. <x1, .., xn>denotes vector with n components of which are x1, x2, x3, ..,xn. |R| denotes the magnitude of the vector R.
|<a, b>| = magnitude of vector = (a 2+ b 2)
|<x1, .., xn>| = (x12+ .. + xn2)
<a, b> + <c, d> = <a+c, b+d>
<x1, .., xn> + <y1, .., yn>= < x1+y1, .., xn+yn>
k <a, b> = <ka, kb>
k <x1, .., xn> = <k x1, .., k x2>
<a, b> <c, d> = ac + bd
<x1, .., xn> <y1, ..,yn> = x1 y1 + .. + xn yn>
R S= |R| |S| cos ( = angle betweenthem)
R S= S R
(a R) (bS) = (ab) R S
R (S + T)= R S+ R T
R R = |R| 2
|R x S| = |R| |S| sin ( = angle betweenboth vectors). Direction of R x S is perpendicularto A & B and according to the right hand rule.
| i j k | R x S = | r1 r2 r3 | = / |r2 r3| |r3 r1| |r1 r2| \ | s1 s2 s3 | \ |s2 s3| , |s3 s1| , |s1 s2| /
S x R = - R x S
(a R) x S = R x (a S) = a (Rx S)
R x (S + T) = R x S + Rx T
R x R = 0
If a, b, c = angles between the unit vectors i, j,k and R Then the direction cosines are set by:
cos a = (R i) / |R|; cos b = (R j) / |R|; cos c = (R k) / |R|
|R x S| = Area of parrallagram with sides Rand S.
Component of R in the direction of S = |R|cos = (R S) / |S|(scalar result)
Projection of R in the direction of S = |R|cos = (R S) S/ |S| 2 (vector result)