Last time, we considered how to represent algebraically the division of a line segment in a given ratio. At the end, we touched on a subject I recalled discussing extensively almost four years ago: that such a “division” can be either internal (inside the segment, as you’d expect) or external (elsewhere on the line containing …
A series of recent questions dealt with proportional division of a line segment. The context was vectors, and we’ll use them a lot, though the main ideas can be understood using ordinary geometry. We’ll see a mistake so easy to make that AI did it just as humans do; and how textbooks can make it …
(A new question of the week) We were recently asked to check work on an interesting little question about parallel vectors, and I was almost convinced that there was no solution … until I realized there was one! How was it missed? How can we avoid doing that? That’s our goal today.
(A new question of the week) Here we have a different kind of question than usual: A conjecture about distances between points, with a request for confirmation. Normally we like to just give hints to help a student figure something out; this was a request for a theorem that ought to exist, and trying to …
A recent question about the resultant velocity of an airplane illustrates different ways to make a diagram showing the bearings of air velocity and wind velocity, and to work out angles without getting too dizzy.
(A new question of the week) Among many interesting recent questions we have one about vectors and equations in three dimensions. We’ll see four different ways to find the distance from a point to a line, proving two formulas and catching some of the errors one can make along the way. We’ll also see a …
(An archive question of the week) We’ve looked at the scalar (dot) product and the vector (cross) product; but there is one answer in the Ask Dr. Math archives that was too long to fit in either post. Here we’ll see again where the two familiar products come from, while looking deeper into the math …
Last time, we looked at the scalar, or dot, product of vectors, focusing on proving the equivalence of two ways to define it. This time, we’ll look at the vector, or cross, product in the same way. The distinction between dot and cross product reflects the symbol used, u · v vs. u × v, …
We’re looking at the concept of vectors at an introductory level. Last week we looked at how they are defined in this context (as quantities with magnitude and direction), and how they are added (which is really part of the definition). Our collection of answers from Ask Dr. Math this time focuses on the ideas …
Because we have had a number of questions about vectors recently, I thought it might be time to look at various facets of that topic. Here, we will start with some ideas about what vectors, and their most basic operations, are. Next week, we’ll get into the far more interesting topic of multiplying vectors.