continued from
>>14301067We now have
This shows that
when . To prove the case where |B| = 0, simply note that both sides are 0.
Let be the angle between A and B.
When k > 0, , and , we have
.
When k < 0, , and , we have
.
When either k, |A|, or |B| are zero, then
.
By the commutativity of the dot project, we also have
and
.
Since the basis vectors i, j, k are perpendicular with length 1, we have
and
.
We can then write any two vectors in terms of their components and expand the product:
