To determine the magnitude of the vector \(a + b\), we will first break down each vector into its component form, calculate the resulting vector's components, and then find its magnitude. Let's go through this step-by-step.
### Step 1: Convert Angles to Radians
Given the angles in degrees:
- Angle of vector \(a\) is \(25^\circ\)
- Angle of vector \(b\) is \(105^\circ\)
The angles are converted to radians using the formula:
[tex]\[ \text{Angle in radians} = \text{Angle in degrees} \times \frac{\pi}{180} \][/tex]
### Step 2: Calculate the x and y Components of Each Vector
- For vector \(a\):
[tex]\[
a_x = |a| \cos(\theta_a)
\][/tex]
[tex]\[
a_y = |a| \sin(\theta_a)
\][/tex]
Where \(|a| = 12\) and \(\theta_a = 25^\circ\).
- For vector \(b\):
[tex]\[
b_x = |b| \cos(\theta_b)
\][/tex]
[tex]\[
b_y = |b| \sin(\theta_b)
\][/tex]
Where \(|b| = 7\) and \(\theta_b = 105^\circ\).
Given the x and y components:
- \(a_x = 10.8757\)
- \(a_y = 5.0714\)
- \(b_x = -1.8117\)
- \(b_y = 6.7615\)
### Step 3: Sum the x and y Components to Find Resultant Vector
[tex]\[
\text{Resultant}_x = a_x + b_x
\][/tex]
[tex]\[
\text{Resultant}_y = a_y + b_y
\][/tex]
Using the given components:
[tex]\[
\text{Resultant}_x = 10.8757 + (-1.8117) = 9.064
\][/tex]
[tex]\[
\text{Resultant}_y = 5.0714 + 6.7615 = 11.8329
\][/tex]
### Step 4: Calculate the Magnitude of the Resultant Vector
The magnitude of the resultant vector can be found using the Pythagorean theorem:
[tex]\[
|a + b| = \sqrt{\text{Resultant}_x^2 + \text{Resultant}_y^2}
\][/tex]
Using the resultant components:
[tex]\[
|a + b| = \sqrt{(9.064)^2 + (11.8329)^2}
\][/tex]
Given the result for the magnitude:
[tex]\[
|a + b| = 14.9
\][/tex]
### Conclusion
Thus, the magnitude of the vector \(a + b\), rounded to the nearest decimal, is \(14.9\).
The correct answer is:
[tex]\[ 14.9 \][/tex]
