Question

let |a| = 12 at an angle of 25° and |b| = 7 at an angle of 105°. what is the magnitude of a + b? round to the nearest decimal.
12.8
14.9
15.4
17.3

Explanation:

Step1: Find the components of vectors

For vector \(a\) with \(|a| = 12\) and \(\theta_a=25^{\circ}\):
\(a_x = |a|\cos\theta_a=12\cos25^{\circ}\approx12\times0.9063 = 10.8756\), \(a_y=|a|\sin\theta_a = 12\sin25^{\circ}\approx12\times0.4226=5.0712\)
For vector \(b\) with \(|b| = 7\) and \(\theta_b = 105^{\circ}\):
\(b_x=|b|\cos\theta_b=7\cos105^{\circ}\approx7\times(- 0.2588)=-1.8116\), \(b_y=|b|\sin\theta_b=7\sin105^{\circ}\approx7\times0.9659 = 6.7613\)

Step2: Calculate the components of \(a + b\)

\((a + b)_x=a_x + b_x=10.8756-1.8116 = 9.064\)
\((a + b)_y=a_y + b_y=5.0712 + 6.7613=11.8325\)

Step3: Find the magnitude of \(a + b\)

Using the formula \(|a + b|=\sqrt{(a + b)_x^{2}+(a + b)_y^{2}}\)
\(|a + b|=\sqrt{9.064^{2}+11.8325^{2}}=\sqrt{82.155 + 140.017}=\sqrt{222.172}\approx14.9\)

Answer:

14.9