Question

find the direction of the vector sum \\(\vec{a} + \vec{b}\\).
vector a is 1.23 m long in a \\(42.9^\circ\\) direction.
vector b is 5.58 m long in a \\(32.5^\circ\\) direction.

Explanation:

Step1: Resolve Vector A into components

To find the x - component of vector \(\vec{A}\), we use the formula \(A_x = A\cos\theta_A\), where \(A = 1.23\space m\) and \(\theta_A=42.9^{\circ}\).
\(A_x=1.23\cos(42.9^{\circ})\approx1.23\times0.733\approx0.902\space m\)
To find the y - component of vector \(\vec{A}\), we use the formula \(A_y = A\sin\theta_A\)
\(A_y = 1.23\sin(42.9^{\circ})\approx1.23\times0.679\approx0.835\space m\)

Step2: Resolve Vector B into components

To find the x - component of vector \(\vec{B}\), we use the formula \(B_x = B\cos\theta_B\), where \(B = 5.58\space m\) and \(\theta_B = 32.5^{\circ}\)
\(B_x=5.58\cos(32.5^{\circ})\approx5.58\times0.843\approx4.704\space m\)
To find the y - component of vector \(\vec{B}\), we use the formula \(B_y = B\sin\theta_B\)
\(B_y=5.58\sin(32.5^{\circ})\approx5.58\times0.537\approx2.996\space m\)

Step3: Find the components of the resultant vector \(\vec{R}=\vec{A}+\vec{B}\)

The x - component of the resultant vector \(R_x=A_x + B_x\)
\(R_x=0.902+4.704 = 5.606\space m\)
The y - component of the resultant vector \(R_y=A_y + B_y\)
\(R_y=0.835 + 2.996=3.831\space m\)

Step4: Find the direction of the resultant vector

The direction \(\theta\) of the resultant vector is given by \(\tan\theta=\frac{R_y}{R_x}\)
\(\tan\theta=\frac{3.831}{5.606}\approx0.683\)
\(\theta=\arctan(0.683)\approx34.3^{\circ}\)

Answer:

The direction of the vector sum \(\vec{A}+\vec{B}\) is approximately \(34.3^{\circ}\)