Distributivity of Dot Product and Cross Product



Coplanar Vectors


When three vectors are coplanar, it means that they lie on the same plane. Let A, B, and C be the three coplanar vectors. Then, we can write:

A = B + C

Taking the dot product of both sides with another vector D, we get:

A · D = (B + C) · D

Using the distributive property of dot product, we get:

A · D = B · D + C · D

Similarly, taking the cross product of A with D, we get:

A x D = (B + C) x D

Using the distributive property of cross product, we get:

A x D = B x D + C x D

Therefore, we have shown that the dot product and cross product are distributive when three vectors are coplanar.

General Case


In the general case, let A, B, and C be any three vectors. Then, we can write:

A = B + C

Taking the dot product of both sides with another vector D, we get:

A · D = (B + C) · D

Using the distributive property of dot product, we get:

A · D = B · D + C · D

However, taking the cross product of A with D does not give a distributive property in the general case. Instead, we get:

A x D = B x D + C x D + (B x C) x D

The additional term (B x C) x D arises due to the non-commutative nature of cross product. Therefore, we have shown that the dot product is distributive in the general case, but the cross product is not.

In summary, the dot product is distributive in both coplanar and general cases, while the cross product is distributive only in coplanar cases.