2. 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.

3. 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) 2 + 1 2 )=  6 n = i + k so |n| =  (1 2 + 0 2 + 1 2 )=  2 m.n = (2 X 1) + (-1 X 0) + (1 X 1)= 2 + 0 + 1= 3 (This is non-calculator !!) cos  = m. n | m || n | = 3.  6 X  2 = 3.  12 = 3.  4 X  3 =  3.. 2  = cos -1 (  3 / 2 ) = 30°=  / 6

4. 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 = ( ) - ( ) 7 0 5 3 2 1 = ( ) 4 -2 4 |PQ| =  (4 2 + (-2) 2 + 4 2 ) =  36 = 6 PR = r – p = ( ) - ( ) 11 2 -5 3 2 1 = ( ) 8 0 -6 |PR| =  (8 2 + 0 2 + (-6) 2 ) =  100 = 10

5. PR.PQ = (4 X 8) + (-2 X 0) + (4 X (-6))= 32 + 0 – 24 = 8 cos  = PR.PQ |PQ||PR| = 8 6 X 10 = 2 / 15  = cos -1 ( 2 / 15 ) = 82.3°