Question #7693

r
and r2
r
are unit vectors in the x-y plane making angles a and b with the positive
x-axis. By considering r1 . r2
r r
, derive
cos (a − b) = cos a cos b + sin a sin b

Expert's answer

Calculate dot product of these vectors in two ways:

1. using formula (r1,r2)=|r1|*|r2|*cos(r1,r2)=1*1*cos(a-b)

2. using coordinates:

since these vectors are radius-vectors and their end points are on the unit circle, coordinates of these vectors are following

r1={cos a,sin a}

r2={cos b,sin b}

So their dot product (r1,r2)=cos a*cos b+sin a*sin b

Therefore

cos(a-b)=cos a*cos b+sin a*sin b

1. using formula (r1,r2)=|r1|*|r2|*cos(r1,r2)=1*1*cos(a-b)

2. using coordinates:

since these vectors are radius-vectors and their end points are on the unit circle, coordinates of these vectors are following

r1={cos a,sin a}

r2={cos b,sin b}

So their dot product (r1,r2)=cos a*cos b+sin a*sin b

Therefore

cos(a-b)=cos a*cos b+sin a*sin b

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