Question

If `A= <8 ,-5 ,6 >` and `B= <7 ,1 ,0 >`, what is `A*B -||A|| ||B||`?

Answer 1

`bb ul(A) * bb ul(B) - || bb ul(A) || \ || bb ul(B) || = 51 - 25sqrt(2)`

Explanation:

We start with the vectors `A` and `B`

`bb ul(A) = << 8,-5,6 >>`
`bb ul(B) = << 7,1,0 >>`

The dot (or scalar) product is given by:

`bb ul(A) * bb ul(B) = << 8,-5,6 >> * << 7,1,0 >>`
`\ \ \ \ \ \ \ \ \ = (8)(7) + (-5)(1) + (6)(0)`
`\ \ \ \ \ \ \ \ \ = 56 -5 + 0`
`\ \ \ \ \ \ \ \ \ = 51`

And the norms are given by:

`|| bb ul(A) || = || << 8,-5,6 >> ||`
`\ \ \ \ \ \ \ = sqrt(8^2+(-5)^2+6^2)`
`\ \ \ \ \ \ \ = sqrt(64+25+36)`
`\ \ \ \ \ \ \ = sqrt(125)`
`\ \ \ \ \ \ \ = 5`

`|| bb ul(B) || = || << 7,1,0 >> ||`
`\ \ \ \ \ \ \ = sqrt(7^2+1^2+0^2)`
`\ \ \ \ \ \ \ = sqrt(49+1+0)`
`\ \ \ \ \ \ \ = sqrt(50)`
`\ \ \ \ \ \ \ = 5sqrt(2)`

So then:

`bb ul(A) * bb ul(B) - || bb ul(A) || \ || bb ul(B) || = 51 - 25sqrt(2)`