To find the magnitude and direction of the resultant vector formed by combining vectors with given magnitudes and angles, we will add their components step-by-step. Here's a detailed solution:
### Step 1: Break Down Each Vector into Components
For each vector with magnitude [tex]\(M\)[/tex] and angle [tex]\(\theta\)[/tex] (measured from the positive x-axis), we can determine the x and y components using the following formulas:
- [tex]\( x = M \cdot \cos(\theta) \)[/tex]
- [tex]\( y = M \cdot \sin(\theta) \)[/tex]
Given vectors:
1. Magnitude: [tex]\(8.06\)[/tex], Angle: [tex]\(60.26^\circ\)[/tex]
2. Magnitude: [tex]\(19.42\)[/tex], Angle: [tex]\(55.50^\circ\)[/tex]
3. Magnitude: [tex]\(8.06\)[/tex], Angle: [tex]\(29.74^\circ\)[/tex]
4. Magnitude: [tex]\(19.42\)[/tex], Angle: [tex]\(34.51^\circ\)[/tex]
#### Converting Degrees to Radians
Angles must be converted from degrees to radians since trigonometric functions typically use radians. Here, the conversion is:
[tex]\[ \theta_{\text{rad}} = \theta \cdot \frac{\pi}{180} \][/tex]
### Components Calculation
1. Vector 1:
- [tex]\(x_1 = 8.06 \cdot \cos(60.26^\circ) = 3.998\)[/tex]
- [tex]\(y_1 = 8.06 \cdot \sin(60.26^\circ) = 6.998\)[/tex]
2. Vector 2:
- [tex]\(x_2 = 19.42 \cdot \cos(55.50^\circ) = 11.000\)[/tex]
- [tex]\(y_2 = 19.42 \cdot \sin(55.50^\circ) = 16.005\)[/tex]
3. Vector 3:
- [tex]\(x_3 = 8.06 \cdot \cos(29.74^\circ) = 6.998\)[/tex]
- [tex]\(y_3 = 8.06 \cdot \sin(29.74^\circ) = 3.998\)[/tex]
4. Vector 4:
- [tex]\(x_4 = 19.42 \cdot \cos(34.51^\circ) = 16.003\)[/tex]
- [tex]\(y_4 = 19.42 \cdot \sin(34.51^\circ) = 11.002\)[/tex]
### Step 2: Sum the Components
Sum the x and y components from all four vectors to obtain the resultant vector components.
- [tex]\( x_{\text{total}} = x_1 + x_2 + x_3 + x_4 \)[/tex]
[tex]\[ x_{\text{total}} = 3.998 + 11.000 + 6.998 + 16.003 = 38.000 \][/tex]
- [tex]\( y_{\text{total}} = y_1 + y_2 + y_3 + y_4 \)[/tex]
[tex]\[ y_{\text{total}} = 6.998 + 16.005 + 3.998 + 11.002 = 38.004 \][/tex]
### Step 3: Calculate the Magnitude of the Resultant Vector
[tex]\[ R = \sqrt{x_{\text{total}}^2 + y_{\text{total}}^2} \][/tex]
[tex]\[ R = \sqrt{38.000^2 + 38.004^2} \][/tex]
[tex]\[ R = 53.742 \][/tex]
### Step 4: Calculate the Direction of the Resultant Vector
[tex]\[ \theta_{\text{res}} = \tan^{-1}\left(\frac{y_{\text{total}}}{x_{\text{total}}}\right) \][/tex]
[tex]\[ \theta_{\text{res}} = \tan^{-1}\left(\frac{38.004}{38.000}\right) \][/tex]
[tex]\[ \theta_{\text{res}} = 45.004^\circ \][/tex]
Therefore, the magnitude of the resultant vector is [tex]\(53.742\)[/tex], and its direction is [tex]\(45.004^\circ\)[/tex].
