find the direction of the vector sum \\(\\vec{a} + \\vec{b}\\).\nvector a is 1.23 m long in a…

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})