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>#?

1 Answer
Dec 21, 2016

Answer:

#<<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 #