Find a unit vector orthogonal to u and v

1. Form the cross product of u and v:

 

uxv = (-3i - j + k)x(-3i + 2j + 3k)

 

→ (-3i)x(-3i) + (-3i)x2j + (-3i)x3k 

   + (-j)x(-3i) + (-j)x2j   + (-j)x3k

   + kx(-3i)    +  kx2j     +  kx3k

 

→ 9ixi - 6ixj -9ixk + 3jxi -2jxj - 3jxk - 3kxi + 2kxj + 3kxk

 

Now for cross products we have. ixi = jxj = kxk = 0

                                                     ixj = -jxi = k

                                                     jxk = -kxj = i

                                                     kxi = -ixk = j

 

So: uxv → 0 - 6k + 9j -3k - 0 - 3i - 3j - 2i + 0 

 

      uxv → -5i + 6j - 9k

 

2. To get the unit vector divide each coefficient by sqrt(5^2 + 6^2 + 9^2) → sqrt(142)

 

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