Question

If `A= <-5 ,3 ,-3 >` and `B= <2 ,2 ,-7 >`, what is `A*B -||A|| ||B||`?

Answer 1

`bb ul A * bb ul B - || bb ul A || \ || bb ul B || = 17 - sqrt(43)sqrt(57)`

Explanation:

We have:

`bb ul A = << -5, 3, -3 >>` and `bb ul B = << 2, 2, -7 >>`

And so we compute the Scalar (or dot product):

`bb ul A * bb ul B= << -5, 3, -3 >> * << 2, 2, -7 >>`

`\ \ \ \ \ \ \ \ \ = (-5)(2) + (3)(6) + (-3)(-7)`

`\ \ \ \ \ \ \ \ \ = -10 + 6 + 21`

`\ \ \ \ \ \ \ \ \ = 17`

And we compute the vector norms (or magnitudes):

`|| bb ul A || = || << -5, 3, -3 >> ||`

`\ \ \ \ \ \ \ = sqrt( << -5, 3, -3 >> * << -5, 3, -3 >> )`

`\ \ \ \ \ \ \ = sqrt( (-5)^2+ (3)^2 + (-3)^2 )`

`\ \ \ \ \ \ \ = sqrt( 25 + 9 + 9)`

`\ \ \ \ \ \ \ = sqrt( 43 )`

Similarly,

`|| bb ul B || = || << 2, 2, -7 >> ||`

`\ \ \ \ \ \ \ = sqrt( << 2, 2, -7 >> * << 2, 2, -7 >> )`

`\ \ \ \ \ \ \ = sqrt( (2)^2+ (2)^2 + (-7)^2 )`

`\ \ \ \ \ \ \ = sqrt( 4+4+49)`

`\ \ \ \ \ \ \ = sqrt( 57 )`

So that:

`bb ul A * bb ul B - || bb ul A || \ || bb ul B || = 17 - sqrt(43)sqrt(57)`