Question

Find the unit vector in the same direction as $\mathit{v}$.

role="math" localid="1646752725885" $\mathit{v}=5\mathit{i}$

Short answer

The unit vector in the same direction as$\mathit{v}$is $\mathit{i}$.

Step-by-step solution

Step 1. Given information.

Consider the given question,

$\mathit{v}=5\mathit{i}$

The magnitude of the vector $\mathit{v}=a_1\mathit{i}+b_1\mathit{j}$,

$\left|\left|\mathit{v}\right|\right|=\sqrt{{\left(a_1\right)}^2+{\left(b_1\right)}^2}$

Compare $\mathit{v}=5\mathit{i}$with the standard form $\mathit{v}=a_1\mathit{i}+b_1\mathit{j}$,

$$\begin{gathered} a_1=5, \\ {b}_1=0 \end{gathered}$$

Thus,

$$\begin{gathered} \left|\left|\mathit{v}\right|\right|=\sqrt{5^2+0^2} \\ \left|\left|\mathit{v}\right|\right|=5 \end{gathered}$$

Step 2. Find the unit vector.

Consider the given question,

The unit vector in the same direction as any non-zero vector $\mathit{v}=a_1\mathit{i}+b_1\mathit{j}$,

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

Substitute the values in the above equation,

$$\begin{gathered} \mathit{u}=\frac{5\mathit{i}+0\mathit{j}}{5} \\ \mathit{u}=\mathit{i} \end{gathered}$$

Therefore, the unit vector in the direction of$\mathit{v}$is$\mathit{i}$.