Linear Algebra and Its Applications, Exercise 3.1.6

Exercise 3.1.6. What vectors are orthogonal to $(1, 1, 1)$ and $(1, -1, 0)$ in $\mathbb{R}^3$ ? From these vectors create a set of three orthonormal vectors (mutually orthogonal with unit length).

Answer: If $x = (x_1, x_2, x_3)$ is a vector orthogonal to both $(1, 1, 1)$ and $(1, -1, 0)$ then the inner product of $x$ with both vectors must be zero. This corresponds to the system $Ax = 0$ where

$A = \begin{bmatrix} 1&1&1 \\ 1&-1&0 \end{bmatrix}$

To solve the system we perform Gaussian elimination. We start by subtracting 1 times row 1 from row 2:

$\begin{bmatrix} 1&1&1 \\ 1&-1&0 \end{bmatrix} \Rightarrow \begin{bmatrix} 1&1&1 \\ 0&-2&-1 \end{bmatrix}$

The echelon matrix has 2 pivots in columns 1 and 2, so $x_1$ and $x_2$ are basic variables and $x_3$ is a free variable.

Setting $x_3 = 1$ from row 2 we have $-2x_2 -x_3 = -2x_2 - 1 = 0$ or $x_2 = -\frac{1}{2}$ . From row 1 we have $x_1 + x_2 + x_3 = x_1 - \frac{1}{2} + 1 = 0$ or $x_1 = -\frac{1}{2}$ . So the vector $(-\frac{1}{2}, -\frac{1}{2}, 1)$ is a solution to the system and thus a vector orthogonal to $(1, 1, 1)$ and $(1, -1, 0)$ .

Note that scalar multiples of the vector $(-\frac{1}{2}, -\frac{1}{2}, 1)$ are all orthogonal to $(1, 1, 1)$ and $(1, -1, 0)$ . Also note that the vectors $(1, 1, 1)$ and $(1, -1, 0)$ are orthogonal to one another.

To produce orthonormal vectors we can take the vectors above and divide them by their lengths. The length of $(1, 1, 1)$ is $\sqrt{1^2+1^2+1^2} = \sqrt{3}$ , the length of $(1, -1, 0)$ is $\sqrt{1^2 + (-1)^2} = \sqrt{2}$ , and the length of $(-\frac{1}{2}, -\frac{1}{2}, 1)$ is

$\sqrt{(-\frac{1}{2})^2 + (-\frac{1}{2})^2 + 1^2} = \sqrt{\frac{3}{2}}$

The three orthonormal vectors are then as follows:

$\begin{bmatrix} \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}} \end{bmatrix} \qquad \begin{bmatrix} \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \\ 0 \end{bmatrix} \qquad \begin{bmatrix} -\frac{1}{\sqrt{6}} \\ -\frac{1}{\sqrt{6}} \\ \frac{2}{\sqrt{6}} \end{bmatrix}$

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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Linear Algebra and Its Applications, Exercise 3.1.6

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