Question

Let It denote the plane parallel to the xz-plane through the point (1, 2, 3). Let pi_2 denote the plane x - y + z = 1. (a) Show that these two planes intersect in a line, and find an equation for this line.

Answer 1

To show that the planes intersect in a line, find the direction vector by taking the cross product of the normal vectors. Choose a point that satisfies both equations. Write the equation of the line using the direction vector and a point on the line.

Step-by-step explanation:

To show that the two planes intersect in a line, we need to find the direction vector of this line. The direction vector can be found by taking the cross product of the normal vectors of the two planes. The normal vector of Let It is (0, 1, 0) and the normal vector of pi_2 is (1, -1, 1). Taking the cross product of these two vectors, we get (-1, -1, -1).

Next, we need to find a point on the line. Since the line lies on both planes, we can choose any point that satisfies both equations. Substituting the coordinates of (1, 2, 3) into the equation x - y + z = 1, we get 1 - 2 + 3 = 1. Therefore, (1, 2, 3) is a point on the line.

Finally, we can write an equation for the line using the direction vector (-1, -1, -1) and the point (1, 2, 3):

x = 1 - t

y = 2 - t

z = 3 - t

where t is a parameter representing the distance along the line.

Learn more about Planes and lines