Question

Use the component method to add the $\overrightarrow{A}$ vectors and $\overrightarrow{B}$ shown in Figure P3.11. Both vectors have magnitudes of$3.00m$and vector $\overrightarrow{A}$ makes an angle of with the x axis. Express the resultant $\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}$ in unit-vector notation.

Short answer

$\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}=(2.6\hat{\mathrm{i}}+4.5\hat{\mathrm{j}})\mathrm{m}$

Step-by-step solution

Define the components

In a two-dimensional coordinate system, the x-component andy-component are commonly considered to be the components of a vector. It can be written as_{$V=\left(vx,vy\right)$}with V denoting the vector.

These are the components of vectors created along the axes. In this article, we shall find the components of any given vector using formulas for both two-dimensional and three-dimensional coordinate systems.

$$\begin{gathered} v_x=V\cos \theta \\ {v}_y=V\sin \theta \end{gathered}$$

State the given data

$$\begin{gathered} A=3\text{m} \\ B=3\text{m} \\ {\theta }_A=30° \\ {\theta }_B=90° \end{gathered}$$

Find the component vector A

Discover the following components to express A in component form:

$$\begin{gathered} A_x=A\cos {\theta }_A \\ =3\cos \left(30°\right) \\ =2.6m \end{gathered}$$

$$\begin{gathered} A_y=A\cos {\theta }_A \\ =3\cos \left(30°\right) \\ =1.5m \end{gathered}$$

$$\begin{gathered} \overrightarrow{\mathrm{A}}={\mathrm{A}}_{\mathrm{x}}\hat{\mathrm{i}}+{\mathrm{A}}_{\mathrm{y}}\hat{\mathrm{j}} \\ =(2.6\hat{\mathrm{i}}+1.5\hat{\mathrm{j}})\mathrm{m} \end{gathered}$$

Find the component vector B

Discover the following components to express B in component form

$$\begin{gathered} B_x=0\text{m} \\ {B}_y=3\text{m} \\ \overrightarrow{\mathrm{B}}={\mathrm{B}}_{\mathrm{x}}\hat{\mathrm{i}}+{\mathrm{B}}_{\mathrm{y}}\hat{\mathrm{j}} \\ =\left(3\hat{\mathrm{j}}\right)\mathrm{m} \end{gathered}$$

Now find the resultant of A and B.We will add the components.

_{$$\begin{gathered} \overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}=(2.6\hat{\mathrm{i}}+1.5\hat{\mathrm{j}})\mathrm{m}+\left(3\hat{\mathrm{j}}\right)\mathrm{m} \\ =(2.6\hat{\mathrm{i}}+4.5\hat{\mathrm{j}})\mathrm{m} \end{gathered}$$}

The resultant vector is$\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}=(2.6\hat{\mathrm{i}}+4.5\hat{\mathrm{j}})\mathrm{m}$