# Finding points of intersection using vectors

• jaejoon89
In summary, the lines L1 and L2 do not have any points of intersection since the system of equations for their coordinates has no solution.
jaejoon89
Find the points of intersection of the lines...

L1: R = 2i + 3j + 3k + t(i - 2j + 5k)
L2: (x + 3) / 2 = (y + 1) / 2 = -z

(I assume the plural in points is wrong... since that would be impossible)
R, i, j, k are vectors; x, y, z are not

---

x = 2 + t
y = 3 - 2t
z = 3 + 5t
for some value(s) of t. If this point also lies on L2, then (x + 3)/2 = -z, so that:
(2 + t + 3)/2 = -(3 + 5t)
5 + t = -6 - 10t
11t = -11
t = -1
For this to really be on L2, you also need to have (y + 1)/2 = -z, or:
(3 - 2t + 1)/2 = -(3 + 5t)
4 - 2t = -6 - 10t
8t = -10
t = -5/4

The two t's don't equal each other. What am I doing wrong?

I parametrized the equations for L2 using a different parameter: w.

x = 2w - 3
y = 2w - 1
z = -w

For any point (x, y, z) to lie on L1 and L2, it must be that the x coordinates on both lines are equal as must be the y and z coordinates.

So,
2 + t = 2w + 3
3 - 2t = 2w - 1
3 + 5t = -w

These are three equations in two unknowns, an overdetermined system.

I ended up with no solution, which means (assuming my calculations are correct) that the two lines don't intersect.

## 1. How do you find the point of intersection between two vectors?

To find the point of intersection between two vectors, you first need to set up the equations of the two lines represented by the vectors. Then, you can use substitution or elimination to solve for the variables and find the coordinates of the point of intersection.

## 2. What is the significance of finding points of intersection using vectors?

Finding points of intersection using vectors is important in many fields of science, such as physics and engineering. It allows us to determine where two lines or planes intersect, and this information can be used to solve problems related to motion, forces, and geometry.

## 3. Can vectors be used to find points of intersection in three-dimensional space?

Yes, vectors can be used to find points of intersection in three-dimensional space. In this case, we would need to set up three equations (one for each dimension) and solve for the three variables to find the coordinates of the point of intersection.

## 4. Is it possible for two vectors to have no point of intersection?

Yes, it is possible for two vectors to have no point of intersection. This occurs when the two lines represented by the vectors are parallel and never intersect.

## 5. Are there any limitations to using vectors to find points of intersection?

One limitation of using vectors to find points of intersection is that it only works for linear equations. It cannot be used for non-linear equations, such as circles or parabolas. Additionally, it may be more challenging to find the point of intersection if the equations are not in standard form.

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