Let's solve the problem step by step: ### **Step 1: Extract Given Information** - Magnitudes: - \(|\vec{A}| = 8.5\) - \(|\vec{B}| = 3.5\) - \(|\vec{C}| = 9.0\) - Angles with x-axis: - \(\theta_A = 61^\circ\) - \(\theta_B = 71^\circ\) - \(\theta_C = 62^\circ\) --- ### **Step 2: Find Cartesian Components** For a vector \(\vec{V}\) with magnitude \(V\) and angle \(\theta\) from the x-axis: - \(V_x = V \cos\theta\) - \(V_y = V \sin\theta\) --- #### **Vector \(\vec{A}\):** - \(A_x = 8.5 \cos(61^\circ)\) - \(A_y = 8.5 \sin(61^\circ)\) #### **Vector \(\vec{B}\):** - \(B_x = 3.5 \cos(71^\circ)\) - \(B_y = 3.5 \sin(71^\circ)\) #### **Vector \(\vec{C}\):** - \(C_x = 9.0 \cos(62^\circ)\) - \(C_y = 9.0 \sin(62^\circ)\) --- ### **Step 3: Calculate the Components** #### \(\vec{A}\): - \(A_x = 8.5 \times \cos(61^\circ) \approx 8.5 \times 0.4848 = 4.1208\) - \(A_y = 8.5 \times \sin(61^\circ) \approx 8.5 \times 0.8746 = 7.4341\) #### \(\vec{B}\): - \(B_x = 3.5 \times \cos(71^\circ) \approx 3.5 \times 0.3256 = 1.1396\) - \(B_y = 3.5 \times \sin(71^\circ) \approx 3.5 \times 0.9455 = 3.3092\) #### \(\vec{C}\): - \(C_x = 9.0 \times \cos(62^\circ) \approx 9.0 \times 0.4695 = 4.2255\) - \(C_y = 9.0 \times \sin(62^\circ) \approx 9.0 \times 0.8829 = 7.9461\) --- ### **Step 4: Write Vectors in Cartesian Components** \[ \vec{A} = (4.12, 7.43) \] \[ \vec{B} = (1.14, 3.31) \] \[ \vec{C} = (4.23, 7.95) \] --- ### **Step 5: Find the X Component of \(\vec{D}\)** Given: \[ \vec{D} = -1.0 \vec{A} + 3.0 \vec{B} + \vec{C} \] The x-component is: \[ D_x = -1.0 A_x + 3.0 B_x + C_x \] \[ D_x = -1.0(4.1208) + 3.0(1.1396) + 4.2255 \] \[ D_x = -4.1208 + 3.4188 + 4.2255 \] \[ D_x = (-4.1208 + 3.4188) + 4.2255 = (-0.702) + 4.2255 = 3.5235 \] --- ### **Final Answers** #### **a) Components of Vectors** - \(\vec{A} = (4.12, 7.43)\) - \(\vec{B} = (1.14, 3.31)\) - \(\vec{C} = (4.23, 7.95)\) #### **b) X Component of \(\vec{D}\)** \[ \boxed{3.52} \] (Rounded to 2 decimal places)

The components of a vector are determined by projecting its magnitude onto the x and y axes using trigonometric functions based on the angle it makes with the x-axis. This allows vectors to be expressed in Cartesian coordinates, facilitating vector addition and other operations.

**Final Answers:** **a) Components of the three vectors:** - \(\vec{A} \approx (4.12, 7.43)\) - \(\vec{B} \approx (1.14, 3.31)\) - \(\vec{C} \approx (4.23, 7.95)\) **b) The x component of \(\vec{D}\):** \[ \boxed{3.52} \]