Angle Between Vectors Given that a. b = | a || b |cos   a b then it must follow that cos  = a. b | a || b | … and this allows us to find the angle.

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Presentation transcript:

Angle Between Vectors Given that a. b = | a || b |cos   a b then it must follow that cos  = a. b | a || b | … and this allows us to find the angle between a & b.

ExIfm = 2i - j + k and n = i + k Find the size of the angle between m & n in radians. ******** m = 2i - j + k so |m| =  (2 2 +(-1) )=  6 n = i + k so |n| =  ( )=  2 m.n = (2 X 1) + (-1 X 0) + (1 X 1)= = 3 (This is non-calculator !!) cos  = m. n | m || n | = 3.  6 X  2 = 3.  12 = 3.  4 X  3 =   = cos -1 (  3 / 2 ) = 30°=  / 6

Ex P is (3,2,1), Q is (7,0,5) & R is (11,2,-5). Find the size of QPR. ^ NAB *********** P Q R  PQ = q – p = ( ) - ( ) = ( ) |PQ| =  (4 2 + (-2) ) =  36 = 6 PR = r – p = ( ) - ( ) = ( ) |PR| =  ( (-6) 2 ) =  100 = 10

PR.PQ = (4 X 8) + (-2 X 0) + (4 X (-6))= – 24 = 8 cos  = PR.PQ |PQ||PR| = 8 6 X 10 = 2 / 15  = cos -1 ( 2 / 15 ) = 82.3°