Final answer:
To find the equation of the line of intersection of two planes, we can first find the direction vector of the line by taking the cross product of the normal vectors of the planes. Then, we can find a point on the line by substituting the values of x, y, and z from one of the planes into the equation of the other plane. Finally, we can write the equation of the line of intersection in vector form using the point and direction vector.
Step-by-step explanation:
To find the equation of the line of intersection of two planes, we can first find the direction vector of the line by taking the cross product of the normal vectors of the planes. The normal vectors of the planes q and r are (-1, 2, -1) and (1, 1, 1), respectively. Taking the cross product of these vectors gives us the direction vector (3, -2, -3).
Next, we can find a point on the line by substituting the values of x, y, and z from one of the planes into the equation of the other plane. Let's use the point (1, 2, -1) from plane q. Substituting these values into the equation of plane r, we get 1 + 2 - 1 = 0, which is true. So (1, 2, -1) lies on the line of intersection.
Finally, we can write the equation of the line of intersection in vector form as r = (1, 2, -1) + t(3, -2, -3), where t is a scalar parameter.