How to find a vector orthogonal to another?

Steps to Solve

1. Identify the Given Vector

Let’s denote the given vector as $\mathbf{v} = \begin{pmatrix} a \ b \ c \end{pmatrix}$, where $a$, $b$, and $c$ are the components of the vector.

2. Choose a Perpendicular Vector

We want to find a vector $\mathbf{u} = \begin{pmatrix} x \ y \ z \end{pmatrix}$ that is orthogonal to $\mathbf{v}$. For two vectors to be orthogonal, their dot product must equal zero.

3. Set Up the Dot Product Equation

The dot product of vectors $\mathbf{u}$ and $\mathbf{v}$ is calculated as: $$ \mathbf{u} \cdot \mathbf{v} = ax + by + cz = 0 $$ This equation will ensure that the two vectors are orthogonal.

4. Solve for One Variable

You can choose values for two of the variables in vector $\mathbf{u}$ (for example, $x$ and $y$) and then solve for $z$. To illustrate, let's say you choose $x = 1$ and $y = 0$, you then find $z$: $$ a(1) + b(0) + cz = 0 \implies z = -\frac{a}{c} \text{ (if } c \neq 0\text{)} $$

5. Specify a Solution

After solving for $z$, you can write vector $\mathbf{u}$ as: $$ \mathbf{u} = \begin{pmatrix} 1 \ 0 \ -\frac{a}{c} \end{pmatrix} $$ This vector $\mathbf{u}$ is now perpendicular to $\mathbf{v}$.