# Basic Linear Algebra Problem

• Yosty22

## 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 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.

## 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?