Question

find the unit vector in the same direction as$v$$v=3i-4j$

Short answer

The unit vector in the direction of the vector$v=3i-4j$is$u=\frac{3}{5}i-\frac{4}{5}j$

Step-by-step solution

Step 1. Given data

The given vector is

$v=3i-4j$

Step 2. Finding the magnitude of the given vector

The magnitude of the vector $v=3i-4j$is

$$\begin{gathered} \left|v\right|=\sqrt{{\left(3\right)}^2+{(-4)}^2} \\ \Rightarrow \left|v\right|=\sqrt{9+16} \\ \Rightarrow \left|v\right|=\sqrt{25} \\ \Rightarrow \left|v\right|=5 \end{gathered}$$

Step 3. Finding the unit vector

The unit vector $u$in the direction of $v$is:

$u=\frac{v}{\left|\left|v\right|\right|}$

Substitute the values of vector and magnitude vector in the formula.

$$\begin{gathered} u=\frac{3i-4j}{\left|5\right|} \\ \Rightarrow u=\frac{3}{5}i-\frac{4}{5}j \end{gathered}$$