Basic Linear Algebra Problem

1. Jan 29, 2015

Yosty22

1. The problem statement, all variables and given/known data

The vectors that are perpendicular to (1,1,1) and (1,2,3) lie on a ____.

2. Relevant equations

3. The attempt at a solution
This is really straight forward, but I cannot validate the answer to myself. The textbook says that they should lie on a line, but why is this? Obviously if a vector, say A = <a,b,c> is perpendicular to (1,1,1) and (1,2,3), A dot (1,1,1) = A dot (1,2,3) = 0. This means a+b+c = a + 2b + 3c, or 2c = -b.

How does this result let you know that any vector A such that A is perpendicular to both (1,1,1) and (1,2,3) has components <a,-2c,c>? Is it because A can be anything and it only depends on the other two components?

2. Jan 29, 2015

MostlyHarmless

Try visualizing it, or holding up two pencils, it really doesnt matter which direction you point them in, there is a unique unit vector that is perpendicular to both. That is there is a unique "direction" in which a vector can point such that, that vector is perpendicular to your two vector. i.e. a line.

3. Jan 29, 2015

slider142

You haven't quite finished. Now that you know that 2c = -b, which means b = -2c, you can replace every occurrence of the variable b with its equivalent value of -2c. This means, for the first equation a + b + c = 0, that a - 2c + c = 0, which allows you to solve for the value of a in terms of the single variable c. What does this imply about the character of A?

4. Jan 29, 2015

Yosty22

So if you know that b = -2c, and you know that a+b+c = 0, then a = c. This means that the vector A is made up of components <c,-2c,c>. So is this saying that since you can describe vector A with just a single variable (c) that is is described by a line?

5. Jan 29, 2015

MostlyHarmless

Kind of, because a vector of that form will be a scalar multiple of the vector <1,-2,1>, but the generalization you made doesn't quite hold true. Could you think pf any examples where a vector is in terms of only one variable but does not describe a line?

6. Jan 30, 2015

Stephen Tashi

Actually there are two unit vectors perpendicular to both but they line on the same line.