Vectors

x+1
___
3

=

y+4
____
-3

=
z
__
2
Ive worked out the two points on this line -1,-4,0 and 2,-7,2 are these correct?

I have two points but when asked find the position vector of line described by the equation which one do i use? or is the vector between those two points?
Thnx

(edited 13 years ago)

Reply 1

ttoby

15

Those points are correct - you can in fact check them for yourself to see if they satisfy (x+1)/3=(y+4)/-3=z/2.

You're looking for a line in the form x=a+tb, where a is a vector on the line (any vector) and b is a vector pointing in the direction of the line. So for instance, you could let a be (-1,-4,0) and b could be (2,-7,2) - (-1,-4,0).

Reply 2

Jowhar

OP

I did this before and i thought it was weird so would
(-1,-4,0) + k(2,-7,2) be ok because i always thought position vector was something from the origin and i dunno in that form in doesnt make sense. And from that how would you find the vector parallel to b?

(edited 13 years ago)

Reply 3

ttoby

15

Original post by Jowhar

I did this before and i thought it was weird so would
(-1,-4,0) + k(2,-7,2) be ok because i always thought position vector was something from the origin and i dunno in that form in doesnt make sense. And from that how would you find the vector parallel to b?

When you have a line in the form x=a+kb then you're basically saying start at the origin then move by position vector a. This puts you on the line. Once you're on the line you can move any distance you like along the line. This is represented by adding kb (where b is a vector parallel to the line). By varying k you can move to various different points on the line. For example see this diagram and the position vectors of the green points:

http://cosketch.com/Saved/ppC3dE5Q

So to answer your question, b is not supposed to be a vector from the origin - it has to be parallel to the line. Using the information you have been given, one way of getting a vector parallel to the line is to get a vector going from one point to another. You can obtain this vector by subtracting the position vector of one point from another, which is what I did. Your method won't work because (2,-7,2) isn't parallel to the line.

To find a vector parallel to b, just use b itself.

Reply 4

ztibor

10

Original post by Jowhar

x+1
___
3

=

y+4
____
-3

=
z
__
2
Ive worked out the two points on this line -1,-4,0 and 2,-7,2 are these correct?

I have two points but when asked find the position vector of line described by the equation which one do i use? or is the vector between those two points?
Thnx

THe vector equation of the line

\displaystyle \vec{r}=\vec{r_0}+k\cdot \vec{d}

where

\vec{r}(x,y,z)

points to the a running point on the line.
For every real value of k it points to a determined point of the line.

\vec{r_0}(x_0,y_0,z_0)

is a given point on the line
(the line passing through it)

\vec{d}(d_1,d_2,d_3)

is the given direction vector
(The line parallel with it)
The scalar equivqlent of this vector equítion is

\displaystyle \frac{x-x_0}{d1}=\frac{y-y_0}{d_2}=\frac{z-z_0}{d_3}

As you see you can read the vectors from this equation.
The coordinates of direction vector - you wrote as b - are in the denominators.
Naturally it is possible to calculate other so type of vectors but
I think it being most simple.

(edited 13 years ago)

Vectors

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