Math2.org Math Tables: Vectors |
(Math) |
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. <x_{1}, .., x_{n}>denotes vector with n components of which are x_{1}, x_{2}, x_{3}, ..,x_{n}. |R| denotes the magnitude of the vector R.
|<a, b>| = magnitude of vector = (a^{ 2}+ b^{ 2})
|<x_{1}, .., x_{n}>| = (x_{1}^{2}+ .. + x_{n}^{2})
<a, b> + <c, d> = <a+c, b+d>
<x_{1}, .., x_{n}> + <y_{1}, .., y_{n}>= < x_{1}+y_{1}, .., x_{n}+y_{n}>
k <a, b> = <ka, kb>
k <x_{1}, .., x_{n}> = <k x_{1}, .., k x_{2}>
<a, b> <c, d> = ac + bd
<x_{1}, .., x_{n}> <y_{1}, ..,y_{n}> = x_{1} y_{1} + .. + x_{n} y_{n}>
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 = | r_{1} r_{2} r_{3} | = / |r_{2} r_{3}| |r_{3} r_{1}| |r_{1} r_{2}| \ | s_{1} s_{2} s_{3} | \ |s_{2} s_{3}| , |s_{3} s_{1}| , |s_{1} s_{2}| /
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 parallelogram 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)