3D Vectors: Components, Magnitude, Direction, Operations, and Unit Vectors, Lecture notes of Mathematics

Download 3D Vectors: Components, Magnitude, Direction, Operations, and Unit Vectors and more Lecture notes Mathematics in PDF only on Docsity! 2.2 VECTORS IN THREE-DIMENSIONAL SPACE The 3D space The set of all ordered triples of real numbers is called as the three-dimensional number space. 3 R ( ){ }∈= z,y,x|z,y,x R Distance and midpoint points: ( )1111 z,y,xP ( )2222 z,y,xP ( ) ( ) ( )222 zzyyxx −+−+− Distance: or 21PP ( )21 P,Pd 121212 Midpoint:       +++ 222 212121 21 zz , yy , xx M PP Exercise. Determine the distance between the given points and the midpoint of the segment joining them. ( )2342 −− ,,P( )4231 ,,P − Vector in 3D A vector in three-dimensional space is an ordered triple of real numbers . a, bc,b,a and c are called components of the vector. x y z α β γ β γ α MUST!!! Initial point: Terminal point: VECTOR COMPONENTS: ( )ttt z,y,x ( )iii z,y,x ititit zz,yy,xx −−− Magnitude and direction Consider vector . c,b,aA = 222 cbaA ++= If , and are the direction α β γ angles, A a cos =α A b cos =β A c cos =γ 1222 =++ γβα coscoscoswhere Solution (continued) 212 ,,A − 3=A A a cos =α A b cos =β A c cos =γ 3 2 =αcos 3 1− =βcos 3 2 =γcos 3 4 5 4 5 y z initial: terminal: ( )354 ,,− ( )542 ,,− 212 ,,A − γ 1 2 3 4 5 1 2 -1-2-3-4-5 -1 -2 -3 -4 1 2 3 -2 -3 -4 -5 x α β Operations on vectors in 3D 321 b,b,bB=321 a,a,aA= SUM: 332211 ba,ba,baBA +++=+ NEGATIVE: 321 a,a,aA −−−=− DIFFERENCE: 332211 ba,ba,baBA −−−=− SCALAR PRODUCT: 321 ca,ca,cacA= Example. Solution: Consider and . Determine the unit vector in the direction of . 322 ,,A −= 024 ,,B = BA 23 − BA 23 − 02423223 ,,,, −−= 048966 ,,,, −−= 094686 −−−−= ,, 9214 ,,−= Solution (continued) BA 23 − 9214 ,,−= BAU 23 − BA BA 23 23 − − = 9214− ,, ( ) 222 9214 ++− = 184196 9214 ++ − = ,, Solution (continued) BAU 23 − 184196 9214 ++ − = ,, 218 9214 ,,− = 218 9 218 2 218 14 ,, − = 610140950 .,.,.=