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.

$$\begin{gathered} \left|\left|\mathit{v}\right|\right|=8, \\ \alpha =45° \end{gathered}$$

Short answer

The vector in the standard form is $4\sqrt{2}\mathit{i}+4\sqrt{2}\mathit{j}$.

Step-by-step solution

Step 1. Given information.

Consider the given question,

$$\begin{gathered} \left|\left|\mathit{v}\right|\right|=8, \\ \alpha =45° \end{gathered}$$

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$\mathit{v}\mathbf{,}\mathit{i}$.

Step 2. Write the vector.

The vector with magnitude $8$and angle $45°$is given below,

role="math" localid="1646755598042" $$\begin{gathered} \mathit{v}=\left|\left|\mathit{v}\right|\right|\left(\cos \alpha \mathit{i}+\sin \alpha \mathit{j}\right) \\ \mathit{v}=8\left(\cos 45°\mathit{i}+\sin 45°\mathit{j}\right) \\ \mathit{v}=8\left(\frac{1}{\sqrt{2}}\mathit{i}+\frac{1}{\sqrt{2}}\mathit{j}\right) \\ \mathit{v}=4\sqrt{2}\mathit{i}+4\sqrt{2}\mathit{j} \end{gathered}$$

Therefore, the vector with magnitude$8$and angle$45°$is$4\sqrt{2}\mathit{i}+4\sqrt{2}\mathit{j}$.