If #A= <8 ,3 ,-1 ># and #B= <6 ,4 ,3 >#, what is #A*B -||A|| ||B||#?

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Answer 1 · 2018-07-01T12:24:54

Dot product is defined in #RR^3# by

#(a_1,a_2,a_3)·(b_1,b_2,b_3)=a_1b_1+a_2b_2+a_3b_3#

By other hand vector module is defined by dot product by itself, it´s say #sqrt(veca·veca)# in the same terms of above.

In our case

#vecA·vecB=8·6+3·4-1·3=57#

#abs(vecA)=sqrt(8^2+3^2+1^2)=sqrt74#

#abs(vecB)=sqrt(6^2+4^2+3^2)=sqrt61#

Finally: #vecA·vecB-abs(vecA)·abs(vecB)=57-sqrt74sqrt61=#

#=57-67.1863=-10.1863#