# Intersection of line through a plane

• goonking
In summary, the intersection of a line and a plane is the point or set of points where the line and the plane meet or cross each other. To find the intersection, you can use the equations of both the line and the plane and solve for the coordinates that satisfy both equations. A line can intersect a plane at more than one point, and if they do not intersect, it means they are either parallel or the line is contained within the plane. This concept is used in various fields such as engineering, architecture, and physics to determine the location of structural elements or calculate the trajectory of moving objects.
goonking

## Homework Statement

Where does the line through (−2, 3, 2) and (3, 5, −1) intersect the plane x + y − 2z = 6?

## The Attempt at a Solution

i used r = r0 + tv

the vector between the 2 given points is <5,2,-3>

r = (-2,3,2) + t<5,2-3>

x = -2 + 5t y=3+2t z=2 - 3t

plugging these into the plane given: -2+5t + 3+2t -2(2-3t)= 6

solving for t , t should equal 9/13 if my algebra is correct.

plugging into the parametric equations , the point of intersection should be (19/13 , 57/13, -1/13)

can someone check all this?

goonking said:

## Homework Statement

Where does the line through (−2, 3, 2) and (3, 5, −1) intersect the plane x + y − 2z = 6?

## The Attempt at a Solution

i used r = r0 + tv

the vector between the 2 given points is <5,2,-3>

r = (-2,3,2) + t<5,2-3>

x = -2 + 5t y=3+2t z=2 - 3t

plugging these into the plane given: -2+5t + 3+2t -2(2-3t)= 6

solving for t , t should equal 9/13 if my algebra is correct.

plugging into the parametric equations , the point of intersection should be (19/13 , 57/13, -1/13)

can someone check all this?

It is correct. Nice work.

goonking
goonking said:

## Homework Statement

Where does the line through (−2, 3, 2) and (3, 5, −1) intersect the plane x + y − 2z = 6?

## The Attempt at a Solution

i used r = r0 + tv

the vector between the 2 given points is <5,2,-3>

r = (-2,3,2) + t<5,2-3>

x = -2 + 5t y=3+2t z=2 - 3t

plugging these into the plane given: -2+5t + 3+2t -2(2-3t)= 6

solving for t , t should equal 9/13 if my algebra is correct.

plugging into the parametric equations , the point of intersection should be (19/13 , 57/13, -1/13)

can someone check all this?
It's not that hard to check it for yourself, and it's a good habit to get into. You can do this by verifying that the point you found is on both the line and on the plane.

To verify that the point is on the line, find the value of t so that 19/13 = -2 + 5t, 57/13 = 3 + 2t, and -1/13 = 2 - 3t. The same value of t should work in all three equations.

To verify that the same point is on the plane, confirm that 19/13 + 57/13 - 2(-1/13) = 6.

RJLiberator and goonking

## 1. What is the intersection of a line and a plane?

The intersection of a line and a plane is the point or set of points where the line and the plane meet or cross each other.

## 2. How do you find the intersection of a line and a plane?

To find the intersection of a line and a plane, you can use the equation of the line and the equation of the plane and solve for the variables that satisfy both equations. The resulting values will give you the coordinates of the intersection point.

## 3. Can a line intersect a plane at more than one point?

Yes, a line can intersect a plane at more than one point. This can occur when the line is parallel to the plane and lies within the plane, or when the line is perpendicular to the plane and passes through the plane at different points.

## 4. What does it mean if a line and a plane do not intersect?

If a line and a plane do not intersect, it means that they are either parallel or that the line is contained within the plane. In both cases, there is no point of intersection between the two.

## 5. How is the intersection of a line and a plane used in real life?

The concept of the intersection of a line and a plane is used in various fields such as engineering, architecture, and physics. For example, in architecture, the concept is used to determine the location of walls, beams, and other structural elements that intersect with each other. In physics, the intersection of a line and a plane can be used to calculate the trajectory of an object in motion as it passes through different planes.

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