Question

Write the vector v in the form $a\mathit{i}+b\mathit{j}$, given its magnitude $\left|\left|\mathit{v}\right|\right|$and the angle $\alpha$ it makes with the positive x-axis.

$$\begin{gathered} \left|\left|\mathit{v}\right|\right|=5, \\ \alpha =60° \end{gathered}$$

Short answer

The vector in the standard form is $\frac{5}{2}\mathit{i}+\frac{5\sqrt{\mathit{3}}}{\mathit{2}}\mathit{j}$.

Step-by-step solution

Step 1. Given information.

Consider the given question,

$$\begin{gathered} \left|\left|\mathit{v}\right|\right|=5, \\ \alpha =60° \end{gathered}$$

The vector with magnitude $\left|\left|\mathit{v}\right|\right|$and the angle $\alpha$is given below,

$\mathit{v}=\left|\left|\mathit{v}\right|\right|\left(\cos \alpha \mathit{i}+\sin \alpha \mathit{j}\right)$

Here, $\alpha$is the angle between $\mathit{v},\mathit{i}$.

Step 2. 

The vector with magnitude $5$and angle $60°$is given below,

$$\begin{gathered} \mathit{v}=5\left(\cos 60°\mathit{i}+\sin 60°\mathit{j}\right) \\ \mathit{v}=5\left(\frac{1}{2}\mathit{i}\mathit{+}\frac{\sqrt{\mathit{3}}}{\mathit{2}}\mathit{j}\right) \\ \mathit{v}=\frac{5}{2}\mathit{i}+\frac{5\sqrt{\mathit{3}}}{\mathit{2}}\mathit{j} \end{gathered}$$

Therefore, the vector with magnitude $5$and angle $60°$is $\frac{5}{2}\mathit{i}+\frac{5\sqrt{\mathit{3}}}{\mathit{2}}\mathit{j}$.