Chapter 14.2, Problem 25WE

To determine

To find: A unit vector orthogonal to $v$ .

Answer to Problem 25WE

The unit vector orthogonal to $v$ is $\frac{4}{5}i+\frac{3}{5}j$ .

Explanation of Solution

Given information:

There is given a vector $\overrightarrow{v}=3i-4j$ .

Let the vector $\overrightarrow{u}=xi+yj$ and consider the given vector $\overrightarrow{v}=3i-4j$ .The dot product of two orthogonal vector is $0$ . It implies,

  $\begin{array}{c}\overrightarrow{u}.\overrightarrow{v}=3x-4y\\ =0\end{array}$

Now, the unit vector orthogonal to vector $v$ can be calculated as,

  $\begin{array}{c}\frac{u}{\parallel u\parallel }=\frac{4i+3j}{\sqrt{{\left(4\right)}^2+{\left(3\right)}^2}}\\ \frac{u}{\parallel u\parallel }=\frac{4}{5}i+\frac{3}{5}j\end{array}$

Therefore, the unit vector orthogonal to $v$ is $\frac{4}{5}i+\frac{3}{5}j$ .