Why the vector equation of Line in threedimension is defined that way?

[Muthumanimaran]

In three dimesnion, equation of a Line is determined using a fixed point on the line and any arbitrary point on the line and position vectors of these two points be r0 and r, then the equation of line is given by r-r0=S (by triangle law of addition), and this S is written as scalar times a vector which is parallel to it( let it be tV) (and I know if two vectors are parallel we can represent one vector as scalar times of another vector) and get the equation of Line r=r0=tV

But what is that S vector and to find the direction of S vector why do we need a new vector( let it be V) which is parallel to this S vector? and how do we know this new vector V is parallel to S vector?

 

[tiny-tim]

Welcome to PF!

Hi Muthumanimaran! Welcome to PF!

I'd put it this way:

to define a line, choose any two points on the line, A and B (and let the origin be O)

then a general point P on the line has AP parallel to AB, so we can write AP = tAB,

so OP = OA + AP

= OA + tAB​

ie r = r_o + tV (where V = AB)

 

[HallsofIvy]

In three dimesnion, equation of a Line is determined using a fixed point on the line and any arbitrary point on the line and position vectors of these two points be r0 and r, then the equation of line is given by r-r0=S (by triangle law of addition), and this S is written as scalar times a vector which is parallel to it( let it be tV) (and I know if two vectors are parallel we can represent one vector as scalar times of another vector) and get the equation of Line r=r0=tV

This is, I assume, a typo- you mean r= r0+ tV.

But what is that S vector and to find the direction of S vector why do we need a new vector( let it be V) which is parallel to this S vector? and how do we know this new vector V is parallel to S vector?

?? We "know this new vector V is parallel to S vector" because it is defined that way: "S is written as scalar times a vector which is parallel to it( let it be tV)".

As for "what is that S vector", it is exactly that you say, r- r0, and, so can be thought of as the vector from r0 to r and so a vector pointing in the direction of the line.

You don't really need a new vector "V". The point is that the length of the vector S is not relevant- any two points on the line would give a vector "S" in the same direction but with different lengths. Some texts might want to define V to be a unit vector but, again, the length doesn't matter.

 

[Muthumanimaran]

Thank you tiny-tim

 

[Abhilash H N]

When we have to define a line, we can do it if can give all points on that line. But the process is difficult because there are infinitely many points on a line. So we can give an equation so as to generalise the points on the line. The equation is nothing but r=r₀+tV.
When we are giving the above equation we are just saying that "you go to the point r₀ are start moving along the direction of the vector V, you will get all the points, in other words the line required". Thats why you need the parameter tV.