Proving that this equation goes through the points

[unknown101]

Homework Statement

Use a vector solution to show that a scalar equation for the line through the points P1(x1,y1) and P2(x2,y2) is y-y1/x-x2=y2-y1/x2-x1

Homework Equations

Find a vector which is normal to the line and then use the dot product of this vector and P1P

The Attempt at a Solution

I tried using numbers but I got lost. I don't what exactly this question is asking. Would I have to find the slope and then come up with a solution from that?

 

[tiny-tim]

Hi unknown101!

Use a vector solution to show that a scalar equation for the line through the points P1(x1,y1) and P2(x2,y2) is y-y1/x-x2=y2-y1/x2-x1

(is that the right answer?)

Hint: call a point on the line P.

What do you know about the vectors PP₁ and P₁P₂ ?

 

[unknown101]

OK so I call a point P. I don't know what you mean about those two vectors. Would I have to find the m which is the slop and then n which is the normal point. Looking at my notes none of this makes sense

 

[tiny-tim]

vector maths

OK so I call a point P. I don't know what you mean about those two vectors. Would I have to find the m which is the slop and then n which is the normal point. Looking at my notes none of this makes sense

ah … doesn't look as if your teacher has introduced you to the joys of vector maths, as opposed to coordinate maths.

Vector maths tries to avoid using coordinates …

for example, (a + b)² = a² + 2a.b + b² is proved by simple algebra, and is a lot easier than the coordinate proof!

In vector maths, you can only really use three combinations …

a.b, axb, and ka (where k is a constant) …

which one(s) do you think would help in proving that PP₁ is parallel to P₁P₂ ?

 

[unknown101]

Would I use a.b to prove that is it parallel?

 

[tiny-tim]

It's easier to use a = kb

 

[unknown101]

So if I use a=kb will I have to find the constant. Looking at the equation I'm trying to prove, do I have to to find out what y and x( the x and y without the number)?

 

[tiny-tim]

Hint: write a = kb in coordinates, and see if you can eliminate k.

 

[unknown101]

I wrote a=kb in coordinates. I don't if I did it correctly.
x1y1=k(x2y2)
x1k1/x2y2=k

Is that all i need to do?

 

[tiny-tim]

x1y1=k(x2y2)

you're making it look like one equation …

it's (x1 , y1) = k(x2 , y2), which is two equations …

so write them out and eliminate k

 

[unknown101]

So I did that I got (x1, y1)/x2, y2)=k

 

[unknown101]

So I did that I got (x1, y1)/x2, y2)=k

So this is the final answer?

 

[tiny-tim]

So I did that I got (x1, y1)/x2, y2)=k

uhhh?

that doesn't even make sense …

you can't divide a vector by another vector

write two equations

 

[unknown101]

You said I should write out 2 equations:
(x1,y1)=k(x2,y2)
x1,y1=kx2,ky2
Is that right?

 

[tiny-tim]

x1,y1=kx2,ky2
Is that right?

Yes … but it would be clearer if you wrote it explicitly as two equations …

anyway, now eliminate k

 

[unknown101]

Yes … but it would be clearer if you wrote it explicitly as two equations …

anyway, now eliminate k

By writing as two equations do you mean as in...
1.(x1,y1)
2.(kx2,ky2)

Eliminate k. The only way I can think for eliminating k is
kx2,ky2=0
kx2,ky2/k=o/k
x2,y2=0

I don't know if I'm doing this right.

 

[tiny-tim]

By writing as two equations do you mean as in...
1.(x1,y1)
2.(kx2,ky2)

They aren't equations …

an equation is something with an = sign in the middle

kx2,ky2=0
kx2,ky2/k=o/k
x2,y2=0

?? this doesn't make any sense at all

write (x1 , y1) = k(x2 , y2) as two equations …

that's two completely separate sentences, each with an = in the middle

 

[unknown101]

Ok I kind of understand it now.
This is what I have so far
x1,y1=kx2,ky2
(x1,y1)/(k,k)=(kx2,ky2)/(k,k)
(x1,y1)/(k,k)=(x2,y2)

 

[unknown101]

Ok I kind of understand it now.
This is what I have so far
x1,y1=kx2,ky2
(x1,y1)/(k,k)=(kx2,ky2)/(k,k)
(x1,y1)/(k,k)=(x2,y2)

Is that right?