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# Find the parametric equation for a line through a point and parallel to two planes

We say that a line is parallel to a plane if the direction vector of the line is parallel to the plane. Let be the line containing the point and parallel to the planes

Find a vector parametric equation for .

The normal vectors of the planes are and . So, the direction vector of will be perpendicular to both of these,

From the first equation we have . Plugging this into the second equation we obtain , which then gives us . Since is arbitrary, we take to obtain a direction vector . Therefore, the vector parametric equation for the line is

# Determine which points are on a plane given by a parametric equation

Let be a plane defined by the scalar parametric equations

1. Determine which of the following points are on .
2. Find vectors such that .

1. The point is not on since

Then,

But then the first coordinate fails since

The point is on since the system of equations

has a solution . Therefore, for .

The point is on since the system of equations

has a solution .

2. Since is on the plane we take . Then from the parametric equations we get the vectors and as the coordinates of and . So, and . Then we have

# Determine parametric equations for given planes

Determine scalar parametric equations for the following planes.

1. The plane through spanned by and .
2. The plane through .

1. The plane through spanned by and is the set of points

Therefore, we have the scalar parametric equation

2. The plane through is the set of points

Therefore, we have the scalar parametric equation