Question

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

Answer 1

`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`