Question

Question #30114

Answer 1

`bb(ul A) * bb(ul B) = 24`

Explanation:

Part (a)

If we have generic vectors:

`bb(ul A) = a bb(ul hat i) + b bb(ul hat j) + c bb(ul hat k)`
`bb(ul B) = d bb(ul hat i) + e bb(ul hat j) + f bb(ul hat k)`

Then the cross product can be calculated by expanding the determinant about the top row:

`bb(ul A) xx bb(ul B) = | ( bb(ul hat i), bb(ul hat j), bb(ul hat k) ), (a,b,c), (d,e,f) |`

resulting in a vector that is perpendicular to both `bb(ul A)` and `bb(ul B)`

Part (b)

We have:

`bb(ul A) = 4bb(ul hat i) + 2 bb(ul hat j) -3bb(ul hat k)`
`bb(ul B) = 3 bb(ul hat i) -4 bb(ul hat k)`

And so the scalar product is:

`bb(ul A) * bb(ul B) = ( 4bb(ul hat i) + 2 bb(ul hat j) -3bb(ul hat k) ) * ( 3 bb(ul hat i) -4 bb(ul hat k) )`

`\ \ \ \ \ \ \ \ \ = (4)(3) + (2)(0) + (-3)(-4)`
`\ \ \ \ \ \ \ \ \ = 12 + 0 +12`
`\ \ \ \ \ \ \ \ \ = 24`