symmetric equations for the line of intersection

Please see the attached file for detailed solution.

Find symmetric equations for a line
Find symmetric equations for the line of intersection of the planes and find the angle between the planes.

x - 2y + z = 1, 2x + y + z = 1

(a) First a point on the line must be found. By setting z = 0, we will find the point where the line intersects the x-y plane.
x - 2y = 1
2x + y = 1
Solve the above equations to find the intersection point in xy plane:

Therefore, the line intersects the xy plane at the point

The two normal vectors for the planes are <1, -2, 1> and <2, 1, 1>. The cross product of the two vectors will give a vector that is parallel to the line of intersection:

Thus, the symmetric equations for the line of intersection are:

(b) the angle of intersection between the two planes can be found using the equation:

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