Question

If `A= <4, 0 ,-3 >` and `B= <-2 ,5 ,7 >`, what is `A*B -||A|| ||B||`?

Answer 1

The answer is `=-29-5sqrt78=-53.2`

Explanation:

Let `vecA=〈a_1,a_2,a_3〉`

and `vecB=〈b_1,b_2,b_3〉`

The dot product is

`vecA.vecB=〈a_1,a_2,a_3〉.〈b_1,b_2,b_3〉`

`=a_1b_1+a_2b_2+a_3b_3`

and the modulus is

`∥vecA∥=sqrt(a_1^2+a_2^2+a_3^2)`

`∥vecB∥=sqrt(b_1^2+b_2^2+b_3^2)`

Here, `vecA=〈4,0,-3〉`

and `vecB=〈-2,5,7〉`

`vecA.vecB=-8+0-21=-29`

`∥vecA∥=sqrt(16+0+9)=sqrt25=5`

`∥vecB∥=sqrt(4+25+49)=sqrt78`

Therefore,

`vecA.vecB-∥vecA∥∥vecB∥=-29-5sqrt78=-53.2`