Question

If `A= <5 ,1 ,-2 >` and `B= <4 ,-2 ,3 >`, what is `A*B -||A|| ||B||`?

Answer 1

`12-sqrt(870)`

Explanation:

For two vectors `A` and `B` of the form `A=< A_x,A_y,A_z >` and `B=< B_x,B_y,B_z >,` the dot product `A*B` is given by `(a_x*b_x)+(a_y*b_y)+(a_z*b_z)`.

For `A=< 5,1,-2 >` and `B=< 4,-2,3 >`, we have:

`A*B=(5*4)+(1*-2)+(-2*3)=20-2-6=color(blue)12`

The next part is the product of the magnitudes of vectors `A` and `B`. The magnitude of a vector `A=< A_x,A_y,A_z >` is given by:

`|A|=sqrt((A_x)^2+(A_y)^2+(A_z)^2 )`

For the given vectors `A` and `B`:

`|A|=sqrt((5)^2+(1)^2+(-2)^2)=sqrt(25+1+4)=sqrt(30)`

`|B|=sqrt((4)^2+(-2)^2+(3)^2)=sqrt(16+4+9)=sqrt(29)`

We then have `|A|*|B|=sqrt(30)*sqrt(29)=sqrt(870)`

Then `A*B-|A||B|=12-sqrt(870)`.