Definition: The dot product of two vectors multiplies matching components and adds the results, producing a single number that equals the product of lengths times the cosine of the angle between them.
How is it computed?
For u = <a, b> and v = <c, d>, the dot product is u · v = ac + bd. The answer is a number, not a vector. Geometrically, u · v = |u||v| cos(theta), where theta is the angle between them.
What can it tell you?
If u · v = 0, the vectors are perpendicular. A positive result means an acute angle; a negative one, an obtuse angle.
Worked example: u = <2, 5> and v = <7, -1>. Then u · v = 14 – 5 = 9. The lengths are the square root of 29 and the square root of 50, so cos(theta) = 9 / the square root of 1450, about 0.236. The angle is about 76.3 degrees.
Perpendicular test: <6, 2> · <-1, 3> = -6 + 6 = 0.
What is a common error?
Writing a vector answer such as <14, -5>. Mathbot says to add the products, which collapses everything into one number.