Question

If `A= <7 ,-5 ,1 >` and `B= <28 ,6 ,7 >`, what is `A*B -||A|| ||B||`?

Answer 1

`173-5sqrt(2607)`

Explanation:

For a vector `vec X=((x_1),(x_2),(x_3))`, its magnitude is `||vec X||=sqrt(x_1^2+x_2^2+x_3^2)`. For two vectors `vec X=((x_1),(x_2),(x_3))` and `vec Y=((y_1),(y_2),(y_3))`, their dot product is `vec X*vec Y=x_1y_1+x_2y_2+x_3y_3`.

From the above, `vec A*vec B=7*28+ -5*6+1*7=173`. The magnitude of `vec A=||vec A||=sqrt(7^2+(-5)^2+1^2)=sqrt(75)=5sqrt(3)`. The magnitude of `vec B=||vec B||=sqrt(28^2+6^2+7^2)=sqrt(869)`.

Therefore, `vec A*vec B-||vec A||||vec B||=173-5sqrt(3)sqrt(869)=173-5sqrt(2607)`.