How can the line passing through two points be represented by a vector equation?

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In summary, the conversation discusses finding the vector equation for a line passing through two given points, A and B. After attempting various equations and parameters, it is determined that the equation of the line can be represented as R = sA + tB, where s and t must follow the relationship s + t = 1. Manipulating this relationship, the vector equation can be simplified to R = B(1-s) + sA = Bt + sA.
  • #1
EngageEngage
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Homework Statement


Let A and B be the two points with position vectors A and B. Show that the line passing through thezse points may be represented by the vector equation:

R = sA + tB


Homework Equations


R = Ro + tV
where Ro is a point on the line and t is some scalar, and V is a vector pointing in the direction of the line.


The Attempt at a Solution



I have tried writing R = A + t(A - B)
and R = B + s(A - B), and manipulating the equations, however, i don't find the solution they are looking for. Could someone please help me get started on this? thank you
 
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  • #2
It would help a lot if you said what s and t are! One thing I note is that they cannot be independent parameters- two parameters would give the equation of a plane. I notice that if s= 0 and t= 1, then R= B. And if s= 1 and t= 0, R= A. In order that this give a line the equation relating s and t must be linear. Now, what linear function, s= at+ b, gives s= 0 when t= 1 and s= 1 when t= 0? I think there is a good change that when you get the answer you will say, "Oh, of course!"
 
  • #3
s + t = 1 is the relationship i suppose, however, I cannot use this to show the desired relationship. Is it a matter of manipulating the equations i came up with, or should I try something else? Also, thank you for the help
 
  • #4
Okay, so you know R = A + t(A - B) is the equation of the line, right? Play around with this and see if you can't get it to look something like something*A + something*B.
 
  • #5
R = B + sA - sB = B(1-s)+ sA = Bt + sA.
Indeed, thanks a lot for all the help!
 
  • #6
EngageEngage said:
s + t = 1 is the relationship i suppose, however, I cannot use this to show the desired relationship. Is it a matter of manipulating the equations i came up with, or should I try something else? Also, thank you for the help
Would the "manipulation" going from s+ t= 1 to s= 1- t be too difficult? Replace s by 1- t in your equation and see what happens.
 

Related to How can the line passing through two points be represented by a vector equation?

What is the meaning of "R = sA + tB"?

The equation "R = sA + tB" represents a vector equation, where R is a vector that is a combination of two other vectors, A and B. The scalars s and t represent the magnitudes of A and B, respectively.

How do you interpret the vectors A and B in this equation?

The vectors A and B represent the direction and magnitude of a quantity. A vector can be represented by an arrow, where the length represents the magnitude and the direction represents the direction of the vector.

What does the scalar s represent in this equation?

The scalar s represents the magnitude of vector A, which is multiplied by the vector A to determine the overall magnitude and direction of vector R. The scalar s can also be thought of as a scaling factor for vector A.

What is the purpose of the vector equation "R = sA + tB"?

The vector equation "R = sA + tB" is used to express the relationship between multiple vectors and their magnitudes. It allows us to combine and manipulate vectors to find the overall magnitude and direction of a quantity.

How is the vector equation "R = sA + tB" used in scientific research?

The vector equation "R = sA + tB" is used extensively in many scientific fields, such as physics, engineering, and mathematics. It allows researchers to analyze and understand the relationships between different quantities and their magnitudes, and to make predictions and calculations based on these relationships.

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