Question

How do you find the cross product and verify that the resulting vectors are perpendicular to the given vectors `<3,2,0>times<1,4,0>`?

Answer 1

`<<0,0,10>>`

Explanation:

We can use the determinant notation to compute the cross product:
`\ \ \ \ \ ( (3),(2),(0) ) xx ( (1),(4),(0) ) = | (ul(hat(i)),ul(hat(j)),ul(hat(k))), (3,2,0),(1,4,0) |`

`:. ( (3),(2),(0) ) xx ( (1),(4),(0) ) = | (2,0),(4,0) | ul(hat(i)) - | (3,0),(1,0) | ul(hat(j)) +| (3,2),(1,4) | ul(hat(k))`

`:. ( (3),(2),(0) ) xx ( (1),(4),(0) ) = (0-0) ul(hat(i)) - (0-0) ul(hat(j)) +(12-2) ul(hat(k))`

`:. ( (3),(2),(0) ) xx ( (1),(4),(0) ) = 0 ul(hat(i)) +0 ul(hat(j)) +10 ul(hat(k))`
`:. ( (3),(2),(0) ) xx ( (1),(4),(0) ) = ( (0),(0),(10) )`, or `<<0,0,10>>`

To confirm that this vector is perpendicular we can check the dot product is zero:

`<<3,2,0>> * <<0,0,10>> = 0+0+0 = 0`
`<<1,4,0>> * <<0,0,10>> = 0+0+0 = 0`