 In geometry, we say two lines are perpendicular to each other. In higher mathematics, we use the term orthogonal. So when are vectors orthogonal? Well, let's see if we can find an orthogonal vector. Suppose we have a vector 5 negative 3. Let's find a vector v perpendicular to u. Then find the dot product. So remember, the vector can be interpreted as the directions for getting from the origin to the point with the same coordinates. So the vector tells us how to get from the origin to the point 5 negative 3. Now the line between the two points has slope negative 3 fifths. So a perpendicular line has slope 5 thirds. Since the vector will tell us how to get from the origin to a point, we want to go from the origin to some point on a line with slope 5 thirds. So a line through the origin with slope 5 thirds passes through the point 3 5. And so a vector running in the same direction will be 3 5. And so we have our vector and a vector perpendicular. We'll find the dot product. And based on one example, we conclude the dot product of orthogonal vectors is 0. So if you're a politician, one example is enough. But for the rest of us we might want a little evidence. So we could try to generalize an example. So let's at least try a vector in R3. Let u be u1 u2 u3. A line going in the same direction has slope uh-oh. We don't have a good definition for slope in more than two dimensions. So suppose vectors u and v are orthogonal. Then in any dimensions we have two sides of a right triangle where the third side is u minus v. So if u and v are orthogonal, then u, v, and u minus v correspond to the three sides of a right triangle. The Pythagorean theorem does apply to higher dimensions. And so we have norm of u squared. Norm v squared is norm u minus v squared. But remember we can express these norms in terms of the dot products. And since the dot product obeys the distributing and associative laws, we have and since we have the dot product of u with itself and the dot product of v with itself we'll just express that as the norm of u squared and the norm of v squared. Substituting these n gives us, and we can simplify, consequently if two vectors are orthogonal their dot product is zero. But notice we can reverse the process. If the dot product of two vectors is zero, then we can multiply by negative two, add u squared plus v squared to both sides. And this expression on the right is the square of the norm of u minus v. And so we get. And since the converse of the Pythagorean theorem is also true, then we can say that if the dot product of two vectors is zero, then the vectors are orthogonal. So let's see how we can use this.