# Equation of a line in 2-space

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• Mar 26th 2011, 05:21 PM
darksoulzero
Equation of a line in 2-space
I have two lines

r: [x,y]=[3,2]+t[4,-5]
q: [x,y]=[1,1]+s[7,k]

a) for what values of k are the lines perpendicular?

I got this by using the dot product of [4,-5].[7,k]=0 and then solving for k.

b) for what value of k are the lines paralell?

I'm not sure how to do this. I know the the normal vector is n=[A,B] and the direction vector is m=[B,-A].

so I took the dot product of [5,4].[7,k] =1
35+4k=1
k=-(34/4)

I made the dot product equal to 1 since cos 0 is 1.
• Mar 26th 2011, 09:46 PM
earboth
Quote:

Originally Posted by darksoulzero
I have two lines

r: [x,y]=[3,2]+t[4,-5]
q: [x,y]=[1,1]+s[7,k]

...
b) for what value of k are the lines paralell?

I'm not sure how to do this. I know the the normal vector is n=[A,B] and the direction vector is m=[B,-A].

so I took the dot product of [5,4].[7,k] =1
35+4k=1
k=-(34/4)

I made the dot product equal to 1 since cos 0 is 1.

Two lines are parallel (in 2-D) if the normal vectors are collinear, that means if

$\displaystyle \overrightarrow{n_1}=c\cdot \overrightarrow{n_2}$

where c is a real constant.

Plug in the known vectors into the equation above. Solve for c and then for k. You should come out with $\displaystyle k = -\frac{35}4$
• Mar 28th 2011, 10:40 AM
alinora11
Re:
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• Mar 28th 2011, 12:25 PM
HallsofIvy
Quote:

Originally Posted by alinora11
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• Mar 29th 2011, 11:50 AM
HallsofIvy
Quote:

Originally Posted by darksoulzero
I have two lines

r: [x,y]=[3,2]+t[4,-5]
q: [x,y]=[1,1]+s[7,k]

a) for what values of k are the lines perpendicular?

I got this by using the dot product of [4,-5].[7,k]=0 and then solving for k.

b) for what value of k are the lines paralell?

I'm not sure how to do this. I know the the normal vector is n=[A,B] and the direction vector is m=[B,-A].

so I took the dot product of [5,4].[7,k] =1
35+4k=1
k=-(34/4)

I made the dot product equal to 1 since cos 0 is 1.

The dot product of two vectors, a and b, is $\displaystyle |a||b|cos(\theta)$. Of course, if the lines are parallel, $\displaystyle cos(\theta)= 1$. If you really want to use the dot product, you would want it equal to $\displaystyle \sqrt{25+ 16}\sqrt{49+ k}$, not 1.

But much simpler is that if two vectors are parallel, one is a multiple of the other. You must have [5,4]= m[7, k] so you have two equations, 5= 7m and 4= mk to solve for the two unknowns. Since you really only want k, write m= 5/7, from the first equation, and replace m by 5/7 in the second equation.