To find the magnitude of the resultant vector [tex]\(\overrightarrow{ A }+\overrightarrow{ B }\)[/tex], we need to break down each vector into its horizontal (x) and vertical (y) components, sum these components, and then determine the magnitude of the resultant vector.
Let's break down the process into detailed steps:
### Step 1: Resolve each vector into its components
We start with the given vectors:
Vector [tex]\(\overrightarrow{A}\)[/tex]:
- Magnitude: [tex]\(2.84 \, \text{m}\)[/tex]
- Direction: [tex]\(23.4^\circ\)[/tex]
Vector [tex]\(\overrightarrow{B}\)[/tex]:
- Magnitude: [tex]\(1.72 \, \text{m}\)[/tex]
- Direction: [tex]\(14.5^\circ\)[/tex]
To find the components, we use trigonometric functions (cosine for the x-component and sine for the y-component).
#### Components of Vector [tex]\(\overrightarrow{A}\)[/tex]
[tex]\[
A_x = 2.84 \cdot \cos(23.4^\circ) = 2.606423136942506
\][/tex]
[tex]\[
A_y = 2.84 \cdot \sin(23.4^\circ) = 1.1279000094027767
\][/tex]
#### Components of Vector [tex]\(\overrightarrow{B}\)[/tex]
[tex]\[
B_x = 1.72 \cdot \cos(14.5^\circ) = 1.6652139414503453
\][/tex]
[tex]\[
B_y = 1.72 \cdot \sin(14.5^\circ) = 0.43065360697363925
\][/tex]
### Step 2: Sum the components of the vectors
Next, we sum the respective components of vectors [tex]\(\overrightarrow{A}\)[/tex] and [tex]\(\overrightarrow{B}\)[/tex]:
[tex]\[
\text{Resultant } x \text{-component}:
\][/tex]
[tex]\[
R_x = A_x + B_x = 2.606423136942506 + 1.6652139414503453 = 4.2716370783928514
\][/tex]
[tex]\[
\text{Resultant } y \text{-component}:
\][/tex]
[tex]\[
R_y = A_y + B_y = 1.1279000094027767 + 0.43065360697363925 = 1.558553616376416
\][/tex]
### Step 3: Calculate the magnitude of the resultant vector
To find the magnitude of the resultant vector [tex]\( \overrightarrow{R} \)[/tex], we use the Pythagorean theorem:
[tex]\[
R = \sqrt{R_x^2 + R_y^2} = \sqrt{4.2716370783928514^2 + 1.558553616376416^2} = 4.54708397817993
\][/tex]
Therefore, the magnitude of the vector sum [tex]\(\overrightarrow{A} + \overrightarrow{B}\)[/tex] is [tex]\(4.54708397817993 \, \text{m}\)[/tex].
