Question

If `A= <2 ,-3 ,9 >` and `B= <-1, 3, 7 >`, what is `A*B -||A|| ||B||`?

Answer 1

`bb(ul(A)) * bb(ul(B)) - || bb(ul(A)) || \ || bb(ul(B)) || = 52 - sqrt(5546)`
`" " = -22.4714710...`

Explanation:

We have:

`bb(ul(A)) = <<2 ,-3 ,9 >>`
`bb(ul(B)) = << -1, 3, 7 >>`

So then we can calculate the scalar product:

`bb(ul(A)) * bb(ul(B)) = <<2 ,-3 ,9 >> * << -1, 3, 7 >>`
`" " = (2)(-1) + (-3)(3) + (9)(7)`
`" " = -2-9+63`
`" " = 52`

And next the metric norms:

`|| bb(ul(A)) || = sqrt( (2)^2+(-3)^2+(9)^2 )`
`" " = sqrt( 4+9+81 )`
`" " = sqrt( 94 )`

`|| bb(ul(B)) || = sqrt( (-1)^2+(3)^2+(7)^2 )`
`" " = sqrt( 1+9+49 )`
`" " = sqrt( 59 )`

So the the result we require is:

`bb(ul(A)) * bb(ul(B)) - || bb(ul(A)) || \ || bb(ul(B)) || = 52 - sqrt(94)sqrt(59)`
`" " = 52 - sqrt(5546)`
`" " = -22.4714710...`