Vectors and the Geometry of Space

Question

Let a=(3,3,6) and b=(9,2,1) be vectors.

Compute the cross product a×b.

a×b = ( , , )

Answer 1

Solution:

The a and b vectors can be written as a^→ = 3i + 3j + 6k , and the b^→ = 9i + 2j + k

a^→ x b^→ = ( 3i x 9i) + ( 3i x 2j) + (3i x k) + (3j x 9i) + ( 3j x 2j) + ( 3j x k) + ( 6k x 9i) + ( 6k x 2j) + ( 6k x k)

note that the cross product values in a three dimensional system i x i = j x j = k x k = 0

also note that by using the right hand rule i x j = k, i x k = -j , j x i = -k , j x k = i , k x i = j , k x j = -i

Applying all this = a^→ x b^→ = 0 + 6k - 3j -27 k + 0 + 3i + 45j -12i + 0 = -9i + 42j -21k

You can also solve this by setting up a matrix i -j k

3 3 6

9 2 1

The solution for this is = [(3)(1 ) -(2)(6)] i - [ (3)(1) -(9)(6)] j + [(3)(2)-(9)(3)] k

solving ( 3 -12) i - ( 3 - 45) j + ( 6 - 27 ) k = -9i + 42j - 21k Same answer as above