The Dot Item MeaningWe specify the dot item of 2 vectorsv = ai + bj as well as w = ci + dj to be v. w = air conditioning + bdNotification that the dot item of 2 vectors is a number as well as not a vector. For 3 dimensional vectors, we specify the dot item likewise:
 Populate Item in R3If v = ai + bj + ck as well as w = di + ej +fkthen v.w = advertisement + be + cf
Instances: If v= 2i + 4j as well as w= i + 5jthen v. w= (2 )( 1) + (4 )( 5) = 22ExerciseLocate the dot item of 2i + j - k andi + 2j The Angle In between 2 VectorsWe specify the angle theta in between 2 vectors v as well as w by the formulav. w cos q=|| v|||| w|| to ensure that
 v. w =|| v|||| w|| cos q
If their angle is an appropriate angle, 2 vectors are called orthogonal. If and also just if v, Wesee that angles are orthogonal. w= 0Example To locate the angle in between v = 2i + 3j + k as well as w= 4i + j + 2k we calculate: and also and also v. w=8+3+2=13Hence Instructions Angles Meaning of Instructions Cosines Allow v=bj+ai+ck be a vector,