Dot Product: Component Form
Definition of the Dot Product
We defined the dot product of two vectors, and , which form an angle of between them as:
Derivation of a New Rule
What if we didn't know the angle between the vectors, but we had their component form? Could we still find their dot product?
Suppose we are given two vectors and . How could we express their component forms? Or, equivalently, how could we find the coordinates of their heads, given that their tails are at the origin?
Recall that a point on the unit circle has coordinates where is in standard position. This corresponds to a unit vector , but they it needs to be scaled to the appropriate magnitude. So using the angles and , we can get that:
Now consider the cosine difference formula:
Multiply both sides by
Notice that the left side matches our definition of dot product, as is the angle between the two vectors and . On the right side, we actually have each of the horizontal and vertical components of vectors and . Rewriting we have:
Dot Product Rule for Component Form
This is the dot product of two vectors given in component form: and
We take the sum of the products of their corresponding components.