7.4. Dot Product and Angle Between Two Vectors http://www.ck12.org
Notice how vectors going in generally the same direction have a positive dot product. Think of two forces acting on
a single object. A positive dot product implies that these forces are working together at least a little bit. Another
way of saying this is the angle between the vectors is less than 90◦.
There are a many important properties related to the dot product that you will prove in the examples, guided practice
and practice problems. The two most important are 1) what happens when a vector has a dot product with itself and
2) what is the dot product of two vectors that are perpendicular to each other.
- v·v=|v|^2
- vanduare perpendicular if and only ifv·u= 0
The dot product can help you determine the angle between two vectors using the following formula. Notice
that in the numerator the dot product is required because each term is a vector. In the denominator only regular
multiplication is required because the magnitude of a vector is just a regular number indicating length.
cosθ=|uu||·vv|Example A
Show the commutative property holds for the dot product between two vectors. In other words, show thatu·v=v·u.
Solution: This proof is for two dimensional vectors although it holds for any dimensional vectors.
Start with the vectors in component form.
u=<u 1 ,u 2 >
v=<v 1 ,v 2 >Then apply the definition of dot product and rearrange the terms. The commutative property is already known for
regular numbers so we can use that.