Finding the intersection of a line and a plane

Steps for finding the intersection of the line and plane

If a line and a plane intersect one another, the intersection will be a single point, or a line (if the line lies in the plane).

To find the point of intersection, we’ll

1. substitute the values of . x. . y. and . z. from the equation of the line into the equation of the plane and solve for the parameter . t.
2. take the value of . t. and plug it back into the equation of the line

This will give us the coordinates of the point of intersection.

The intersection of a line and a plane will either be a single point or a line

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Intersection of a plane and a line given by parametric equations

Example

Find the point where the line intersects the plane.

The line is given by . x=-1+2t. . y=4-5t. and . z=1+t.

The plane is given by . 2x-3y+z=3.

Our first step is to plug the values for . x. . y. and . z. given by the equation of the line into the equation of the plane.

Now we’ll plug the value we found for . t. back into the equation of the line.

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Putting these values together, we can say the point of intersection of the line and the plane is the coordinate point

If we want to double-check ourselves, we can plug this coordinate point back into the equation of the plane.

Since . 3=3. is true, we know that the point we found is a true intersection point with the plane.