Question

8. * Let e1 and e2 be unit vectors in the OXY-plane which make angles α and β with
the x-axis. Show that e1 = i cos α + jsin α, e2 = i cos β + jsin β and prove that
cos(α − β) = cos α cos β + sin α sin β

Answer 1

Answer:

Cos(α − β )  =  CosαCosβ + SinαSinβ

Step-by-step explanation:

e1 is unit vector with α

component along x  axis  = 1Cosα

component along y  axis  = 1Sinα

e1  = iCosα  + JSinα

| e1| = √Cos²α + Sin²α = √1 = 1

e2 is unit vector with β

component along x  axis  = 1Cosβ

component along y  axis  = 1Sinβ

e2  = iCosβ  + JSinβ

| e2| = √Cos²β + Sin²β = √1 = 1

α − β is angle between two vector

Cos(α − β )  =  e1 . e2 / ( |e1 . |e2|

= (iCosα  + JSinα ). (iCosβ  + JSinβ) / (1 * 1)

= (CosαCosβ + SinαSinβ)/1

= CosαCosβ + SinαSinβ

Cos(α − β )  =  CosαCosβ + SinαSinβ