Question

If `A= <7 ,-2 ,1 >` and `B= <-2 ,5 ,7 >`, what is `A*B -||A|| ||B||`?

Answer 1

`bb(ul A) bb( * ) bb(ul B) - || bb(ul A) || \ || bb(ul B) || = -17-18sqrt(13)`

Explanation:

We have:

`bb(ul A) = << 7, -2, 1 >>`
`bb(ul B) = << -2, 5, 7 >>`

So then the scalar product (or dot product) is:

`bb(ul A) bb( * ) bb(ul B) = << 7, -2, 1 >> bb( * ) << -2, 5, 7 >>`
`\ \ \ \ \ \ \ \ \ = (7)(-2) + (-2)(5) + (1)(7)`
`\ \ \ \ \ \ \ \ \ = -14 - 10 + 7`
`\ \ \ \ \ \ \ \ \ = -17`

And the moduli of the vectors are:

`|| bb(ul A) || = sqrt( (7)^2 + (-2)^2 + (1)^2 )`
`\ \ \ \ \ \ \ = sqrt(49+4+1)`
`\ \ \ \ \ \ \ = sqrt(54)`

`|| bb(ul B) || = sqrt( (-2)^2 + (5)^2 + (7)^2 ) =`
`\ \ \ \ \ \ \ = sqrt(4+25+49)`
`\ \ \ \ \ \ \ = sqrt(78)`

And so:

`bb(ul A) bb( * ) bb(ul B) - || bb(ul A) || \ || bb(ul B) || = -17-sqrt(54)sqrt(78)`
`" " = -17-sqrt(4212)`
`" " = -17-18sqrt(13)`