Equation Of A Line In Space

To find the line of intersection of the planes and , we proceed as follows:

Step 1: Find the Direction Vector of the Line

The direction vector of the line of intersection is the cross product of the normal vectors of the two planes.

Normal vector of the first plane .

Normal vector of the second plane .

The cross product is:

Thus, the direction vector is .

Step 2: Find a Point on the Line

To find a point on the line, we can set and solve the system of equations:

From equation (2), . Substitute into equation (1):

Then, . So, the point is .

Step 3: Write the Symmetric Equations of the Line

Using the point and the direction vector , the symmetric equations of the line are: