# Conceptual Understanding in Introductory Physics IX: Solving Dot and Cross Product Equations

The question in this post continues the thread related to vectors. We really need to do a much better job of treating vectors effectively, and accurately, in introductory. I have been heavily influenced by mathematics colleagues, and in particular by Keith Devlin, in that I have come to see that we should focus more on the properties of vectors that allow us to manipulate them mathematically (algebraically really). Students should understand vectors in terms of their commutative, associative, distributive, and linear properties. Fortunately, these properties bring with them a lot of rich and beautiful geometry. It is this beautiful geometry that makes classical vector analysis so useful in the absence of coordinate systems. We need to do more to exploit this!

So here’s the new question:

Consider the vector equation A • B = C (C is a scalar). Consider B and C to be given. (a) Carefully draw a diagram using arrows to represent all possible vectors A that satisfy the equation. In other words, you’re attempting to solve the equation for A and drawing the solution set. (b) What if you now restrict your solution only to vectors A with a fixed (constant) magnitude? Draw arrows representing all such vectors that satisfy the equation. (c) What if you now restrict your solution only to vectors A making a fixed angle with B? Draw arrows representing all such vectors that satisfy the equation. Now consider the vector equation A × B = C (this time C is a vector), with B and C to be given. (d) Carefully draw a diagram using arrows to represent all possible vectors A that satisfy the equation. (e) What if anything, changes if you now specify that all three vectors are mutually orthogonal? Draw arrows representing all such vectors A that satisfy the equation. (f) Revisit the previous question on vector division and answer it again and see if your answer has changed as a result of this question.

Go!

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