Question

If `A= <2 ,-3 ,9 >` and `B= <0 , 3, 7 >`, what is `A*B -||A|| ||B||`?

Answer 1

`bb ul A * bb ul B - || bb ul A || \ || bb ul B || = 54 - sqrt(94)sqrt(58)`

Explanation:

We have:

`bb ul A = << 2, -3, 9 >>` and `bb ul B = << 0, 3, 7 >>`

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

`bb ul A * bb ul B= << 2, -3, 9 >> * << 0, 3, 7 >>`

`\ \ \ \ \ \ \ \ \ = (2)(0) + (-3)(3) + (9)(7)`

`\ \ \ \ \ \ \ \ \ = 0 - 9 + 63`

`\ \ \ \ \ \ \ \ \ = 54`

And we compute the vector norms (or magnitudes):

`|| bb ul A || = || << 2, -3, 9 >> ||`

`\ \ \ \ \ \ \ = sqrt( << 2, -3, 9 >> * << 2, -3, 9 >> )`

`\ \ \ \ \ \ \ = sqrt( (2)^2+ (-3)^2 + (9)^2 )`

`\ \ \ \ \ \ \ = sqrt( 4 + 9 + 81)`

`\ \ \ \ \ \ \ = sqrt( 94 )`

Similarly,

`|| bb ul B || = || << 0, 3, 7 >> ||`

`\ \ \ \ \ \ \ = sqrt( << 0, 3, 7 >> * << 0, 3, 7 >> )`

`\ \ \ \ \ \ \ = sqrt( (0)^2+ (3)^2 + (7)^2 )`

`\ \ \ \ \ \ \ = sqrt( 0+9+49)`

`\ \ \ \ \ \ \ = sqrt( 58 )`

So that:

`bb ul A * bb ul B - || bb ul A || \ || bb ul B || = 54 - sqrt(94)sqrt(58)`