Cross product, is a vector product since it yields another vector rather than a scalar. As with the dot product, the cross product of two vectors contains valuable information about the two vectors themselves.

We may now write the formula for the cross product as,

An equivalent definition of the cross product is,

Find the cross product of a = 3i - 2j - 2k and b = -1i + 5k.

a = 3i - 2j - 2k b = -1i + 5k

Calculate the area of parallelogram spanned by the vectors a = 3i-3j+1k and b = 4i+9j+2k.

a = 3i-3j+1k b = 4i+9j+2k

| a x b | = i( (-3).(2) - (1).(9)) - j((3).(2)-(1).(4)) + k((3).(9) + (3).(4))
| a x b | = -15 i - 2 j + 39 k
The area is | a x b | Using the above expression for the cross product, we find that the area is,
Area = √ (-15)^{2} + (-2)^{2} + 39 ^{2}
Area = 5 √ 70