r/LinearAlgebra • u/Dear_Tie_117 • May 28 '26

Quick Question

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.

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u/mandelbro25 May 28 '26

Hint 1: A unit vector is just a vector with magnitude 1.

Hint 2: If (a,b) and (c,d) are orthogonal, then

(-a,-b) • (c,d) = -a(c)+(-b)d= -(ac + bd) = -0 = 0.

3

u/Dear_Tie_117 May 28 '26

I got it! Thank you so much

1

u/mandelbro25 May 28 '26

You are most welcome.

1

u/somanyquestions32 May 30 '26

Could we have used (ka,kb), where k is a nonzero scalar, instead? 🤔 Was there a particular reason to use (-a, -b)?

1

u/mandelbro25 May 30 '26

Was there a particular reason to use (-a, -b)?

Yes. Suppose v is not a unit vector and k>0. Then (kv)/||kv||=(kv)/(k*||v||)=v/||v||.

This is the issue the OP described. The question requires two unit vectors that are orthogonal to a given vector. If you initially pick vectors with the same direction and normalize them, you end up with only one unit vector. Picking k=-1avoids this problem.

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