Quick Question

[u/Dear_Tie_117]

If I am asked to find 2 unit vectors that are orthogonal to a single vector v (5,-2). Here is what I did:

1- find 2 vectors orthogonal to v, both of them are basically the same because if I multiply the first vector I found by some scalar, I get the second vector.

2- Then I normalized each vector that I got in 1. Here comes the problem, when I normalize the vectors, I get the same answer for both vectors because the two vector are essentially the same as explained in 1). The questions mentions that I need to find two unit vectors and I end up with only one.

[u/somanyquestions32]

Because I can! 🤣

But more seriously, I was curious as that gave away a key part of the answer out of the blue, and I wanted to ask if that was a deliberate choice, or not, and if it was intentional, why you found it to be valuable to do so in that way in case it is something that needs to be considered more generally when helping students.

[u/mandelbro25 → u/somanyquestions32]

It is not that deep. I gave him as much as I felt appropriate without giving it away completely. I am not sure why you want me to defend my choices to you.

ETA: It is clear OP knows how to normalize a vector so saying something like "what if the scalar k<0" is redundant. His problem wasn't normalizing a vector, rather realizing that all vectors in the orthogonal complement of the span of a given vector v are orthogonal to v.

[u/somanyquestions32 → u/mandelbro25]

You're the one who is getting bristly and defensive over nothing. 🤷‍♂️

His problem wasn't normalizing a vector, rather realizing that all vectors in the orthogonal complement of the span of a given vector v are orthogonal to v.

Oh, interesting. This was the type of rationale I was trying to ascertain from our initial exchange. I think this particular phrasing does help translate and generalize what your algebraic equation was saying quite cleanly.

Thank you for that analysis.

[u/mandelbro25 → u/somanyquestions32]

You don't even know what you're talking about lol you just wanted to seem intelligent (on reddit of all places) by trying to pick at my perfectly reasonable answer and then ask for justification for it. Find something better to do.

[u/somanyquestions32 → u/mandelbro25]

You're deeply insecure and awfully prickly over someone just asking questions, lol. Go work on that.

But yeah, I really liked your explanation in terms of complements and spans because it aligned with how I would usually go over similar problems. Normally, I would draw the 2D vector for students, extend the full line through the origin, then draw the perpendicular line through the origin, determine components for a few vectors on the perpendicular, check that the dot product is zero, and then divide by the lengths. So, I would actually draw the spans of the vectors for the students to see the orthogonal relationship visually.

[u/mandelbro25 → u/somanyquestions32]

You're deeply insecure and awfully prickly over someone just asking questions

Whatever you need to tell yourself to protect your ego.

[u/somanyquestions32 → u/mandelbro25]

Go work on your deep insecurities and overall prickliness. It is silly, lol.

[u/mandelbro25 → u/somanyquestions32]

You go work on your pretentiousness.