## Linear Algebra and Its Applications, Exercise 3.4.9

Exercise 3.4.9. Given the three orthonormal vectors $q_1$, $q_2$, and $q_3$, what linear combination of $q_1$ and $q_2$ is the least distance from $q_3$?

Answer: Any linear combination of $q_1$ and $q_2$ is in the plane formed by $q_1$ and $q_2$. The combination closest to $q_3$ is simply the projection of $q_3$ onto that plane. But because $q_3$ is orthogonal to both $q_1$ and $q_2$ it is orthogonal to that plane, and its projection onto the plane is the zero vector. So the linear combination of $q_1$ and $q_2$ closest to $q_3$ is $0 \cdot q_1 + 0 \cdot q_2$.

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, Fifth Edition and the accompanying free online course, and Dr Strang’s other books .

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