We may use the concept of the cross product of vectors to illustrate this process more clearly.

Recall that the cross product U·V of two linearly independent vectors U and V is the vector orthogonal to both of them having length equal to the area of the parallelogram spanned by U and V , and such that U , V and the cross product form a right-handed frame. Then we may express B(t) as T(t) ·P(t) .

Alternatively, we may describe B(t) as the cross product of the velocity and acceleration vectors divided by the length of this cross product.

B=(X^' [!--@char cross--]X^'' )/(X ^'[!--@char cross--]X^{' '})

(Note that the binormal vector is not defined if the velocity and acceleration vectors are linearly dependent, in which case their cross product will be zero). This observation makes it possible for us to define B(t) without reference to the principal normal P(t) .

From this fact, we gain an additional benefit. We may define B(t) first, and then define P(t) to be B(t) ·T(t) . This is the most practical way of obtaining P(t) for vectors in three-space. (For curves of higher dimension, the cross product is not available. In that case, we will use the procedure described in the previous section to define the principal normal.)