## 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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