Linear Algebra and Its Applications, exercise 1.2.12

Exercise 1.2.12. We have two equations, x + y + z = 1 and x + y + z = 2. The first part of the exercise is to sketch the planes in 3-space associated with these two equations; I’m skipping that part. The second part of the exercise is to find a vector that is perpendicular to both of those planes.

Answer: The plane for the equation x + y + z = 1 intersects the x, y, and z axes in the points (1, 0, 0), (0, 1, 0), and (0, 0, 1) respectively; the lines between those three points are in the plane in question. Any vector perpendicular to the plane is therefore going to be perpendicular to each of those three lines.

The line passing through the points (1, 0, 0) and (0, 1, 0) is in the x-y plane and is described by the equation y = -x + 1. One line perpendicular to that line is the z axis, for which x = y = 0. Another line perpendicular to that line is the line y = x in the x-y plane, for which z = 0. These two lines form a plane, and all lines in that plane are also perpendicular to the line between (1, 0, 0) and (0, 1, 0). We also have x = y for all points in the plane. Or to put it another way, the plane is described by the equation x – y = 0.

Similarly, the line passing through the points (0, 1, 0) and (0, 0, 1) is in the y-z plane and is described by the equation z = -y + 1. All lines perpendicular to that line are in the plane for which y = z (or y – z = 0).

Finally, the line passing through the points (1, 0, 0) and (0, 0, 1) is in the x-z plane and is described by the equation z = -x + 1. All lines perpendicular to that line are in the plane for which z = x (or x – z = 0).

For a vector to be perpendicular to the original plane associated with the equation x + y + z = 1, it must be in each of the three planes just described. The intersection of those planes is the line for which x = y = z. So (1, 1, 1) is a vector perpendicular to the original plane.

A similar argument shows that the vector (1, 1, 1) is perpendicular to the plane for which x + y + z = 2.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.

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