Question

If a=8i+j-2k and b=5i-3j+k what is the value of b X a

Answer 1

Distribute the product:

a×b = (8i + j - 2k) × (5i - 3j + k)

a×b = 8i × (5i - 3j + k) + j × (5i - 3j + k) - 2k × (5i - 3j + k)

a×b = (40 (i×i) - 24 (i×j) + 8 (i×k))

… … … + (5 (j×i) - 3 (j×j) + k×j)

… … … + (-10 (k×i) + 6 (k×j) - 2 (k×k))

Recall the definition of the cross product:

i×i = j×j = k×k = 0 (the zero vector)

i×j = k

j×k = i

k×i = j

and for any two vectors x and y, we have x×y = -(y×x).

So we have some cancellation and rewriting:

a×b = (40 (i×i) - 24 (i×j) + 8 (i×k))

… … … + (5 (j×i) - 3 (j×j) + j×k)

… … … + (-10 (k×i) + 6 (k×j) - 2 (k×k))

a×b = (-24k - 8j) + (-5k + i) + (-10j - 6i)

a×b = -5i - 18j - 29k

Then

b×a = -(a×b) = 5i + 18j + 29k