Basic Linear Algebra Problem

[Yosty22]

Homework Statement

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

Homework Equations

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?

Thanks in advance.

 

[MostlyHarmless]

Try visualizing it, or holding up two pencils, it really doesn't 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.

 

[slider142]

Homework Statement

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

Homework Equations

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?

Thanks in advance.

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?

 

[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?

 

[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?

 

[Stephen Tashi]

Try visualizing it, or holding up two pencils, it really doesn't matter which direction you point them in, there is a unique unit vector that is perpendicular to both..

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