Vector geometry - Intersection of lines

• Keshroom
In summary: You will get the coordinates of the point of intersection, which are (3,1/2,-4/3). In summary, to find the coordinates at which two parametric vector equations intersect, equate the components of the vectors and solve for the parameter. Then, substitute the parameter back into either equation to get the coordinates. The coordinates for the intersection of the given equations are (3,1/2,-4/3).
Keshroom

Homework Statement

I have 2 parametric vector equations (of a line)

r(t) = (2,-4,4) + t(1,-3,4)
s(t) = (1,-1,0) + t(2,-1,1)

how do i find the coordinates for which they intersect each other?

Homework Equations

x=a+λv, for some λ in ℝ (parametric vector form of line)

The Attempt at a Solution

As in high school, with the form y=mx+b i would make the 2 equations equal to each other, solve for x, then substitute back into either equations to find y.

I've tried making the (x,y,z) components equal to each other, solve for 't' and substitute back in but i can't get the answer in the back of the book

parametic equations
for r(t): x = 2+t, y= -4-3t, z= 4+4t
for s(t): x = 1+2t, y= -1-t, z= t

Now i did
2+t = 1+2t
t = 1

substituting back into x=2+t: x = 3

i did this for also y and x components and got (3, 1/2, -4/3)
hmmmm
I have a feeling that this method isn't correct :s

Change the notation to
r(t) = (2,-4,4) + t(1,-3,4)
s(u) = (1,-1,0) + u(2,-1,1)
the parameters t and u don't have to be the same for the lines to intersect each other.

Dick said:
Change the notation to
r(t) = (2,-4,4) + t(1,-3,4)
s(u) = (1,-1,0) + u(2,-1,1)
the parameters t and u don't have to be the same for the lines to intersect each other.

alright. Now how do i solve it?

Keshroom said:
alright. Now how do i solve it?

The same way you tried before. Equate components of the vectors and solve them. Try it. The first component gives you 2+t=1+2u. Solve that for t and substitute into the rest.

1. What is the formula for finding the intersection point of two lines?

The formula for finding the intersection point of two lines is as follows:

x = (c2 * b1 - c1 * b2) / (a1 * b2 - a2 * b1)

y = (c1 * a2 - c2 * a1) / (a1 * b2 - a2 * b1)

where a1, b1, c1 are the coefficients of the first line and a2, b2, c2 are the coefficients of the second line.

2. How do you determine if two lines are parallel or intersecting?

If two lines have the same slope, they are parallel and will never intersect. If the slopes are different, the lines will intersect at a single point.

3. Can two lines intersect at more than one point?

No, two lines in a 2-dimensional plane can only intersect at one point. This point is the solution to the system of equations formed by the two lines.

4. What is the significance of the intersection point in vector geometry?

The intersection point of two lines in vector geometry represents the point at which the two lines cross, or the point that they have in common. This point can be used to solve systems of equations, find distances and angles, and determine if lines are parallel or perpendicular.

5. How do you find the intersection point of three or more lines?

To find the intersection point of three or more lines, you can use a method called Gaussian elimination. This involves converting the system of equations into an augmented matrix and using row operations to reduce it to row echelon form. The values in the rightmost column of the matrix will represent the coordinates of the intersection point.

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