Equation of Line: (-1,2,-3) + t(1,-1,-1)

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In summary, the equation for the line passing through the point (-1, 2, -3) in the direction of the vector (1, -1, -1) can be found by using the equation v(t) = v subscript 0 + tv, where v is the direction vector and v subscript 0 is the initial point. By substituting the given values, the equation becomes (-1, 2, -3) + t(1, -1, -1). It is also helpful to understand dot products and cross products to verify the correctness of the equation.
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embury
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Homework Statement



I need to find the equation for the line that passes through the pont (-1, 2, -3) in the direction of the vector (1,-1,-1)

Homework Equations



The equation needs to be in the form v(t)=v subscript 0 + tv

The Attempt at a Solution


I know how to find the equation of a line passing through two points, but I have no idea how to find the equation with only one point heading in the direction of a vector. I'm not really looking for the answer, I'm looking for an explanation on how to find the equation of a line given one point in the direction of a vector. Any help would be appreciated. Thank you.

I think I've found the solution, but I'm not sure. v=(1,-1,-1) and V subscript 0 = (-1,2,-3), therefore the equation of the line is (-1,2,-3) + t(1,-1,-1). Is this right?
 
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  • #2
Take a look at dot products and cross products.
 
  • #3
I'm not sure if your answer is correct, but one way to check it is to use what you already know -- how to find the equation based on two points. Since the line is in the direction of (1,-1,-1) from point (-1,2,-3), then a 2nd point on the line would just be (1,-1,-1) away from the first point, right?
 
  • #4
Thank you. I guess this question isn't that difficult. For some reason I thought it was harder than it was. :blushing: Thank you guys.
 
  • #5
Actually, after the very good response to the "eigenvalue" question, yes, you should be embarrased!:smile:
 

Related to Equation of Line: (-1,2,-3) + t(1,-1,-1)

1. What is the Equation of a Line?

The equation of a line is a mathematical representation of a straight line on a graph. It can be written in the form of y = mx + b, where m is the slope of the line and b is the y-intercept.

2. What does (-1,2,-3) + t(1,-1,-1) represent in the Equation of a Line?

The point (-1,2,-3) is a specific point on the line, and t represents any real number. This is known as the parametric equation of a line, where t is the parameter that determines the position of any point on the line.

3. How can I find the Slope of the Line using this Equation?

To find the slope of the line, we can use the formula m = (y2-y1)/(x2-x1), where (x1,y1) and (x2,y2) are two points on the line. In this case, we can choose any two values of t and plug them into the equation to find the corresponding points, then use the formula to calculate the slope.

4. How many Dimensions does this Equation of a Line have?

This equation represents a line in three dimensions, as it has three variables (x, y, z) and can be graphed on a three-dimensional coordinate plane.

5. Can I use this Equation to find a Specific Point on the Line?

Yes, by choosing a specific value for t, you can plug it into the equation to find the corresponding point on the line. For example, if t = 2, then the point on the line would be (1,1,-5).

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