- Draw and label applicable models with units 2. Vector \( c \) has a magnitude of 6 and makes an angle of \( 15^{\circ} \) with the horizontal, Vector \( d \) has a magnitude of 4 and makes an angle of \( 66^{\circ} \) with the horizontal. What is the magnitude of \( c+ \) d? Round your answer to the nearest hundredth of a unit. *Non-negotiables*: - Combine the horizontal components of each vector, combine vertical components of each vector - Find the magnitude of the resultant vector (the hypotenuse of the newly created right triangle) by using the Pythagorean Theorem - Use calculator and round appropriately

**Step 1. Draw and Label the Vectors** - **Vector \( c \):** - Magnitude: \( 6 \) units - Angle: \( 15^{\circ} \) above the horizontal - Horizontal component: \( c_x=6\cos(15^{\circ}) \) - Vertical component: \( c_y=6\sin(15^{\circ}) \) - **Vector \( d \):** - Magnitude: \( 4 \) units - Angle: \( 66^{\circ} \) above the horizontal - Horizontal component: \( d_x=4\cos(66^{\circ}) \) - Vertical component: \( d_y=4\sin(66^{\circ}) \) *On a diagram, plot a horizontal axis (x-axis) and a vertical axis (y-axis). Draw vector \( c \) starting from the origin, making an angle of \(15^{\circ}\) with the horizontal. Similarly, draw vector \( d \) beginning from the origin with an angle of \(66^{\circ}\) with the horizontal. To add the vectors, place the tail of vector \( d \) at the head of vector \( c \), or equivalently, compute the resultant by adding the components.* --- **Step 2. Combine the Horizontal and Vertical Components** - **Horizontal components:** \[ R_x = c_x + d_x = 6\cos(15^{\circ}) + 4\cos(66^{\circ}) \] - **Vertical components:** \[ R_y = c_y + d_y = 6\sin(15^{\circ}) + 4\sin(66^{\circ}) \] --- **Step 3. Compute the Components Using a Calculator** - Compute \( 6\cos(15^{\circ}) \): \[ 6 \times \cos(15^{\circ}) \approx 6 \times 0.9659 \approx 5.80 \] - Compute \( 4\cos(66^{\circ}) \): \[ 4 \times \cos(66^{\circ}) \approx 4 \times 0.4067 \approx 1.63 \] - **Sum of horizontal components:** \[ R_x \approx 5.80 + 1.63 = 7.43 \] - Compute \( 6\sin(15^{\circ}) \): \[ 6 \times \sin(15^{\circ}) \approx 6 \times 0.2588 \approx 1.55 \] - Compute \( 4\sin(66^{\circ}) \): \[ 4 \times \sin(66^{\circ}) \approx 4 \times 0.9135 \approx 3.65 \] - **Sum of vertical components:** \[ R_y \approx 1.55 + 3.65 = 5.20 \] --- **Step 4. Find the Magnitude of the Resultant Vector** Using the Pythagorean theorem, the magnitude \( R \) of the resultant vector is given by: \[ R = \sqrt{R_x^2 + R_y^2} \] Substitute the computed values: \[ R = \sqrt{(7.43)^2 + (5.20)^2} \approx \sqrt{55.23 + 27.04} \approx \sqrt{82.27} \] \[ R \approx 9.07 \text{ units} \] --- **Final Answer** The magnitude of \( c+d \) is approximately \( 9.07 \) units.