How to find which of three points are on a line?

In summary, P and Q fall on the line because they satisfy the parametric equations while R does not because it does not satisfy the equations. The i and j magnitudes do not affect whether a point falls on the line, but rather all three magnitudes must be compatible with the parametric equations.
  • #1
rocapp
95
0

Homework Statement


Which of these three points falls on the line?

l: r(t)=(i+2j)+t(6i+j-5k)

P(1,2,0); Q(-5,1,5); R(-4,2,5)

I have the answer, but I don't understand why P and Q fall on the line but R does not. Is it because the i and j magnitudes are different for R?

Homework Equations

The Attempt at a Solution



P:
(1+6t) = 1
t=0

2=2+t
t=0

t=0

Q:
-5=1+6t
t=-1

1=2+t
t=-1

t=0

R:
-4=1+6t
t=-5/6

2=t+2
t=0

t=0
 
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  • #2
First, you are right that P and Q fall on the line and R does not. And as you see, R is not on the line because using those points you get 3 incompatible equations in t.

Whether a point falls on a line depends on all 3 magnitudes, i, j and k, and the trivial answer to your question is that most combinations will not work i.e. most 3 dimensional points are not on any given line.

However, working backwards from the equation we can get some insight. For example r(0) = i + 2j. So the point that fits there is P = (1,2,0). When you look at r(-1) = -5i + j - 5k, you see that Q = (-5,1,-5) is a point on the line.

But your incompatible equation for R tells you that there is no possible t for which r(t) = R. It's not really deeper than that.
 
  • #3
Thanks! I think I understand, but what about the t=0 in R? Does this value not count since there is already an incompatible value in its components?
 
  • #4
rocapp said:

Homework Statement


Which of these three points falls on the line?

l: r(t)=(i+2j)+t(6i+j-5k)

P(1,2,0); Q(-5,1,5); R(-4,2,5)

I have the answer, but I don't understand why P and Q fall on the line but R does not. Is it because the i and j magnitudes are different for R?


Homework Equations




The Attempt at a Solution



P:
(1+6t) = 1
t=0

2=2+t
t=0

t=0

Q:
-5=1+6t
t=-1

1=2+t
t=-1

t=0

R:
-4=1+6t
t=-5/6

2=t+2
t=0

t=0

The line has parametric equations x = 1 + 6t, y = 2 + t, z = -5t. A point (a,b,c) lies on the line if the equations a = 1+6t, b = 2+t, c = -5t are compatible; that is, we must get the same t from all three equations.
 
  • #5
Thanks a bunch! That's what I needed.
 

1. How can I determine if three points are on the same line?

In order to determine if three points are on the same line, you can use the slope formula to calculate the slope of the line formed by any two of the three points. If the slope is the same for all three pairs of points, then the points lie on the same line.

2. What is the equation for a line in slope-intercept form?

The equation for a line in slope-intercept form is y = mx + b, where m represents the slope and b represents the y-intercept. This form of the equation is useful for determining if three points lie on the same line, as the slope can be easily calculated from the given points.

3. How do I find the slope of a line given two points?

To find the slope of a line given two points, you can use the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the two points. This will give you the slope of the line passing through those two points.

4. Can I use the distance formula to determine if three points are on the same line?

No, the distance formula is used to find the distance between two points. It cannot be used to determine if three points lie on the same line. Instead, you should use the slope formula to calculate the slope of the line formed by any two of the three points.

5. Are there any special cases where the three points may appear to be on the same line, but are not?

Yes, there are two special cases where the three points may appear to be on the same line, but are not. The first case is when all three points have the same x-coordinate, but different y-coordinates. The second case is when all three points have the same y-coordinate, but different x-coordinates. In both of these cases, the points do not actually lie on the same line.

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