Exercise 3.1.14. Given two vectors and in , show that their difference is orthogonal to their sum if and only if their lengths and are the same.

Answer: First we assume that is orthogonal to . This means that their inner product must be zero, or

So we have which implies that or (keeping in mind that the length of a vector must be a positive quantity).

So if is orthogonal to then we must have .

Now suppose that . This implies that or . We then use the definitions of and to run the argument above in reverse:

Since we see that is orthogonal to .

So if then must be orthogonal to .

Combining the two proofs above, we have shown that for any two vectors and in in , is orthogonal to if and only if

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.