# Help with a relatively simple linear algebra proof

• paulrb
In summary, to show that the lines given by the equations ax + by + c = 0 and bx - ay + d = 0 are perpendicular, one must find a vector in the direction of each line and show that these vectors are orthogonal. This can be done by choosing values for x and y to get two points on the line, and then finding the vector between these points. It is important to note that b cannot equal 0 in order for the equations to be valid.
paulrb

## Homework Statement

Show that the lines given by the equation ax + by + c = 0 and bx - ay + d = 0 (where a, b, c, d are in R) are perpendicular by finding a vector in the direction of each line and showing that these vectors are orthogonal. (Hint: Watch out for the cases in which a or b equals zero.)

## Homework Equations

Vectors a and b are orthogonal if the dot product of a and b are 0.

## The Attempt at a Solution

I have spent a long time trying to figure out this problem but I don't even know how to start. I don't know how to create a vector in the same direction as a line unless I know 2 points on the line or the slope of the line. In this case I don't know either because the equation is so general. That is, unless choosing my own values of a, b, and c is acceptable (Is it?).

I cannot use many techniques to prove this since not much has been introduced in the course. Only basic vector properties, the dot product, and projections have been introduced so far.

Choosing a, b and c is not acceptable. Choosing a value of x or y is. Since those equations are supposed to be true for all x and y. Pick x=0 and x=1. Or y=0 and y=1. Now you have two points.

Thank you, I'm not sure why I didn't think of it like that before...

Sorry, I didn't have time to actually do the problem before. Now that I'm looking at it again, I'm still confused. How do I relate the first equation to the second equation?

For example, if I choose x=0 y=0 for the first equation, what do I do with that? I get c = 0, but that doesn't tell me anything about the second equation. That I can do x=1 y=1 and get c = -(a+b) giving me the vector [1,1] for the first equation but again I don't see what that tells me. I can't just plug in he same values of x and y for the second equation, because obviously that will just give me the same vector.

Sorry if that doesn't make sense...I feel like I'm missing something obvious, but I don't know what it is.

Start with the line ax + by + c = 0. To get a vector in the direction of this line, you need two points. I think this is where you are getting confused.

Take the line y = 3x + 2 as an example. How do you get a point on this line? Well x is the independent variable so you can choose any number. Choose x=0 for simplicity. Then you get that y = 2. So a point on this line is (0,2)

Going back to ax + by + c = 0, you cannot pick x=0 and y=0 at the same time. The value of y depends on what you choose for x (or vice versa). So if you choose x=0, find the y that satisfies the equation a*0 + by + c = 0.

Ok...put the points for y won't be numbers, but variables.

if x = 0, y = -(c/b) giving point (0, -c/b)
if x = 1, y = (-c-a)/b giving point (1, -c/b - a/b)

Now I have a vector: [1, -a/b]

using y = 0 and y = 1 for the second equation will give me [a/b, 1]

These are orthogonal! Thank you!
Now all that is left is the hint part of the problem. b cannot equal 0, or it is invalid. Is there anything special I have to add to account for that?

go back to the equations for the lines ax + by + c = 0 and bx - ay + d = 0. If you substitute b=0 into both equations, what are the lines that do get?

I worked on it a bit more and figured everything out. Thanks for your help.

## What is linear algebra?

Linear algebra is a branch of mathematics that deals with the study of linear equations, matrices, vectors, and vector spaces. It is used to solve problems related to geometry, physics, engineering, and many other fields.

## What is the purpose of a linear algebra proof?

A linear algebra proof is used to demonstrate the validity of a mathematical statement or theorem related to linear algebra. It helps to provide a logical and step-by-step explanation of how the statement is true.

## How can I approach a linear algebra proof?

The best approach to solving a linear algebra proof is to carefully read the question and understand the given information. Then, try to break down the problem into smaller, more manageable steps and use theorems and definitions to guide your reasoning.

## What are some common mistakes to avoid in a linear algebra proof?

Some common mistakes to avoid in a linear algebra proof include not clearly stating the given information, using incorrect notation, skipping steps, and making assumptions without justification. It is important to be thorough and precise in your reasoning.

## Where can I find resources to help with linear algebra proofs?

There are many online resources available for help with linear algebra proofs, such as textbooks, video tutorials, and practice problems. You can also seek assistance from a tutor or your instructor for additional support.

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