# What Vector Spans the Line Defined by These Two Equations?

• MHB
• TheFallen018
In summary, the best way to find a vector that spans the line defined by the given equations is to set y arbitrarily to 1 and solve the resulting system. Another method is to use the fact that both equations have the same coefficient for x, leading to the vector <-15, 1, 3> as a solution.
TheFallen018
Hey,

I've got this problem that I'm trying to work out. I've tried a couple of things, but they don't really get me anywhere.

Here's the problem

Find a vector that spans the line defined by these two equations.
$x+6y+3z=0$
$x+3y+4z=0$

Hi fallen angel,

There are many, many ways to approach this.

The easiest I can think of is the following.
Let's assume that the vector we are searching for has a non-zero y-component.
Then we can set y arbitrarily to 1.
Now solve the system.

Btw, I've picked y to set to 1, since y is the coordinate with the largest coefficient (6), giving us the biggest chance that the other numbers are 'nice' numbers.

I like Serena said:
Hi fallen angel,

There are many, many ways to approach this.

The easiest I can think of is the following.
Let's assume that the vector we are searching for has a non-zero y-component.
Then we can set y arbitrarily to 1.
Now solve the system.

Btw, I've picked y to set to 1, since y is the coordinate with the largest coefficient (6), giving us the biggest chance that the other numbers are 'nice' numbers.
Hey, thanks. That makes sense. I ended up coming up with (-15,1,3), which worked nicely. However, I'm curious about other ways to do this. Is there a fairly systematical method that would allow you to come up with a vector that would contain all values? Something that would look like (-15,1,3)+t(x,y,z).

Thanks

TheFallen018 said:
Hey,

I've got this problem that I'm trying to work out. I've tried a couple of things, but they don't really get me anywhere.

Here's the problem

Find a vector that spans the line defined by these two equations.
$x+6y+3z=0$
$x+3y+4z=0$

The first thing I see is that the two equations start with "x". If we subtract the second equation from the first, we eliminate "x" and get 3y- z= 0. So z= 3y. Putting 3y in for z the two equations become x+ 6y+ 9y= x+ 15y= 0 and x+ 3y+ 12y= x+ 15y= 0. In either case, x= -15y. So <x, y, z>= <-15y, y, 3y>= y<-15, 1, 3>. <-15, 1, 3>, or any multiple of it, spans that line.

## 1. What is a vector that spans two equations?

A vector that spans two equations is a mathematical concept that represents a set of numbers or variables that can satisfy two or more equations simultaneously. In other words, the vector has components that can satisfy both equations at the same time.

## 2. How can I determine if a vector spans two equations?

To determine if a vector spans two equations, you can use the method of elimination. First, solve both equations for a common variable. Then, set the two expressions equal to each other and solve for the remaining variables. If the values of the remaining variables are the same, then the vector spans both equations.

## 3. What is the significance of a vector that spans two equations?

A vector that spans two equations is significant because it represents a solution to a system of equations. This means that it satisfies both equations simultaneously and can be used to find a point of intersection between the two equations.

## 4. Can a vector span more than two equations?

Yes, a vector can span more than two equations. In fact, a vector can span any number of equations as long as there are enough equations to uniquely determine the values of the variables.

## 5. Are there any real-world applications of vectors that span two equations?

Yes, there are several real-world applications of vectors that span two equations. These include finding the intersection of two planes in 3D space, determining the equilibrium point of a system of forces, and solving for the position of an object in motion using multiple equations.

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