L03-Notes-DotProd-KEY+1

6.4 Parallel and Perpendicular Vectors Dot Product : If ± ² ± ² 1 2 1 2 , and , , v v v w w w & )& then the dot product is 1 1 2 2 v w v w v w ³ & )& ± . Ex. Find the dot product for ± ² ± ² 3,4 and 7, 2 . v w ´ ´ ´ & )& ______________________________________________________________________________ Angle between two vectors: To find the angle between two vectors and , v w & )& cos w v w v T )& & ± )& & Ex. Find the measure of the angle between the vectors ± ² ± ² 9, 5 and 7, 1 w v ´ )& & . ______________________________________________________________________________ If cos 0, then 90 T T q so two vectors are perpendicular if their dot product is 0. Two vectors are parallel if one is a multiple of the other. Vectors are said to be orthogonal if they are perpendicular to each other. Ex. Determine whether the given vectors are parallel, perpendicular, or neither. (a) ± ² ± ² 8, 6 , 4, 3 ´ ´ (b) ± ² ± ² 8, 6 , 3, 4 ´ ´ ´ (c) ± ² ± ² 8, 6 , 3, 4 ´ ´ ______________________________________________________________________________ Ex. Given the vectors ± ² ± ² ,3 and 2, 5 k , (a) Determine the value of k so that the given vectors will be parallel. (b) Determine the value of k so that the given vectors will be perpendicular. J W 371 7 4 2 21 8 D J W 9 7 5 i 63 5 58 F F F y xp p É É W 11TH ft Fo cos 0 0.796691 I o cost 0.796691 cost 0 142.8150 4 900 cos 1900 0 IjEw If Ftw parallel p If Yay then J W O T parallel 0 900 8 3 C 6 C 4 I Em yes I a o.o notpa.ae I not perpendicular Neited Iz 3 K E Same scalefactor K z 3 5 0 2K 15 K I 7.5