If #A= <2 ,1 ,-4 ># and #B= <3 ,2 ,7 >#, what is #A*B -||A|| ||B||#?

1 Answer
Y
May 7, 2017

#A*B-|\|A|\||\|B|\|# is asking for the dot product of A and B and the product of the magnitudes of vectors A and B.

For the dot product, we find the sum of the products for the x, y, and z terms, so we find that:
#A*B = (2*3)+(1*2)+(-4*7) = 6+2-28=-20#

For the magnitudes, we use the distance formula for 3D vectors:
#d=sqrt((x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2)#
or in this case, since the vectors already give the values of #x_2-x_1, y_2-y_1, z_2-z_1#, we simply plug in the given values:
#|\|A|\| = sqrt(2^2+1^2+(-4)^2)=sqrt(21)#
#|\|B|\| = sqrt(3^2+2^2+7^2)=sqrt(62)#

By putting all the parts together, we get:
#A*B-|\|A|\||\|B|\|=-20-sqrt(21)*sqrt(62)=-20-sqrt(21*62)=-20-sqrt(1302)~~-56.083#

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