 In geometry, we say two lines are perpendicular to each other. In higher mathematics, we use the term orthogonal, and so we might ask the question, when are two vectors orthogonal? Let's find out. Suppose we have a vector, and we want to find a vector that is perpendicular to it. So remember the vector can be interpreted as the direction for getting from the origin to the point with the same coordinates. So this vector 5-3 tells us how to get from the origin to the point 5-3. Now the line between the two points has slope negative three-fifths, so a perpendicular line has slope five-thirds. Now if I run a line through the origin with slope five-thirds, it will pass through the point three-five, and so a vector running in the same direction as this perpendicular line will be three-five. Now if we find the dot product of these two vectors we get, and based on one example, we conclude the dot product of orthogonal vectors is zero. Now since we're not running for a political office, we probably want to provide evidence, so we could try to generalize an example. So let you be a vector with three components, a line going in the same direction has slope. So while we can prove our theorem using the Pythagorean theorem, that would make a great homework problem or even a problem on a test. Oh well, anyway, with a little more effort and some basic trigonometry we can show the following. The angle theta between two vectors u and v satisfies the equation. The dot product is the product of the norms times the cosine of the angle. So given two vectors, let's find the angle between them. So our theorem says we need to find the dot product and the norms of the two vectors. So our theorem tells us the angle theta satisfies the equation, which we can solve for theta and find. Now suppose the two vectors are orthogonal, or we might say perpendicular. Now let's get used to using the phrase orthogonal. The dot product still satisfies that equation, but if vectors are perpendicular, I mean orthogonal, then our angle is either 90 degrees or 270 degrees, and in either case the cosine of theta is 0. And so we have... Now this is an obvious consequence of a theorem, and in mathematics we call such obvious consequences corollaries. Since we're starting with two vectors that are orthogonal, and we're concluding that their dot product is 0, we can say that if two vectors are orthogonal, then their dot product is 0. Conversely, suppose the dot product of two vectors is 0. So our dot product satisfies the equation, if the dot product is 0 we have, and since this is a product equal to 0, one of the factors must be 0. So either one of the vectors has magnitude 0, or the cosine of theta is 0, which only occurs if the two vectors are perpendicular. Ah, I mean orthogonal. Since vectors with magnitude 0 aren't interesting, we can forbid consideration of such vectors and say that if two non-zero vectors have dot product 0, then they are orthogonal.