Final answer:
To find the intersection point of a line and a plane, draw a sketch, set up a system of equations involving the line's parametric equations and the plane's equation, solve for the parameter, and substitute back to find the intersection coordinates.
Step-by-step explanation:
To find the point where a line intersects a plane, you need to set up a system of equations using both the equation of the line and the equation of the plane. The line is generally given in parametric or vector form, which provides direction ratios. The plane is usually given in the form of an equation Ax + By + Cz + D = 0.
Here is a step-by-step approach to find the intersection:
- Draw a sketch of the problem to visualize the line and the plane and determine the orientation of both entities.
- Identify known and unknown quantities, and identify the system of interest. For instance, if the line is given in parametric form, we have x = x0 + at, y = y0 + bt, z = z0 + ct where (x0, y0, z0) is a point on the line and a, b, c the direction ratios.
- Set up equations by substituting the parametric equations of the line into the equation of the plane. This will give you an equation involving the parameter, usually denoted by t.
- Solve for t. The value of t gives the specific point on the line where it intersects the plane.
- Substitute the value of t back into the line's parametric equations to find the coordinates of the intersection point.
Note that if the line is parallel to the plane (the line's direction vector is orthogonal to the plane's normal vector), there will not be an intersection point, unless the line lies entirely on the plane.