# Basic Linear Algebra Problem

## Homework Statement

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

## 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?

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.

## Homework Statement

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

## 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?

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?

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?

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