Absolutely! Let's write each vector in component form as you requested. The general formula for a vector in the \(xy\)-plane given its magnitude \(r\) and angle \(\theta\) (measured from the positive x-axis) is: \[ \vec{V} = r\cos(\theta) \, \hat{x} + r\sin(\theta) \, \hat{y} \] Let’s use the information: - **Vector A:** magnitude = \(5.0\,\text{cm}\), angle = \(127^\circ\) - **Vector B:** magnitude = \(3.5\,\text{cm}\), angle = \(70^\circ\) - **Vector C:** magnitude = \(7.0\,\text{cm}\), angle = \(20^\circ\) ### Writing Them in Component Form: #### **Vector A** \[ \vec{A} = 5.0\cos(127^\circ)\,\hat{x} + 5.0\sin(127^\circ)\,\hat{y} \] #### **Vector B** \[ \vec{B} = 3.5\cos(70^\circ)\,\hat{x} + 3.5\sin(70^\circ)\,\hat{y} \] #### **Vector C** \[ \vec{C} = 7.0\cos(20^\circ)\,\hat{x} + 7.0\sin(20^\circ)\,\hat{y} \] --- **If you want the numerical values as well:** (rounded to two decimal places) #### **Vector A** - \(\cos(127^\circ) \approx -0.6018\) - \(\sin(127^\circ) \approx 0.7986\) \[ \vec{A} = (5.0 \times -0.6018)\,\hat{x} + (5.0 \times 0.7986)\,\hat{y} = -3.01\,\hat{x} + 4.00\,\hat{y} \] #### **Vector B** - \(\cos(70^\circ) \approx 0.3420\) - \(\sin(70^\circ) \approx 0.9397\) \[ \vec{B} = (3.5 \times 0.3420)\,\hat{x} + (3.5 \times 0.9397)\,\hat{y} = 1.20\,\hat{x} + 3.29\,\hat{y} \] #### **Vector C** - \(\cos(20^\circ) \approx 0.9397\) - \(\sin(20^\circ) \approx 0.3420\) \[ \vec{C} = (7.0 \times 0.9397)\,\hat{x} + (7.0 \times 0.3420)\,\hat{y} = 6.58\,\hat{x} + 2.39\,\hat{y} \] --- ### **Summary Table** | Vector | Magnitude (cm) | Angle (°) | x-component | y-component | |--------|----------------|-----------|-------------|-------------| | A | 5.0 | 127 | -3.01 | 4.00 | | B | 3.5 | 70 | 1.20 | 3.29 | | C | 7.0 | 20 | 6.58 | 2.39 | --- **In standard vector notation:** \[ \boxed{ \begin{align*} \vec{A} &= -3.01\,\hat{x} + 4.00\,\hat{y} \\ \vec{B} &= \;\;1.20\,\hat{x} + 3.29\,\hat{y} \\ \vec{C} &= \;\;6.58\,\hat{x} + 2.39\,\hat{y} \\ \end{align*} } \] Let me know if you need them in a different format!

Certainly! Let's go through each vector step-by-step, including detailed calculations, to convert from magnitude and angle to component form. --- ### **Given Data:** | Vector | Magnitude \( r \) | Angle \( \theta \) (degrees) | |---------|------------------|------------------------------| | **A** | 5.0 cm | 127° | | **B** | 3.5 cm | 70° | | **C** | 7.0 cm | 20° | --- ### **Step 1: Recall the component form formula** \[ \vec{V} = r \cos(\theta) \,\hat{x} + r \sin(\theta) \,\hat{y} \] where: - \( r \) = magnitude of the vector - \( \theta \) = angle in degrees from the positive x-axis --- ### **Step 2: Convert angles from degrees to radians (if necessary)** Since most calculator functions use radians, but many calculators can directly handle degrees, we can work directly in degrees or convert. Conversion: \[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \] But for simplicity, we'll use degrees directly in calculator functions that support degree mode. --- ### **Step 3: Calculate each component** --- ## Vector A - **Magnitude:** \( r_A = 5.0\, \text{cm} \) - **Angle:** \( \theta_A = 127^\circ \) ### Calculate \( \cos(127^\circ) \): \[ \cos(127^\circ) \approx -0.6018 \] ### Calculate \( \sin(127^\circ) \): \[ \sin(127^\circ) \approx 0.7986 \] ### Components: \[ A_x = r_A \times \cos(127^\circ) = 5.0 \times (-0.6018) = -3.009 \approx -3.01\, \text{cm} \] \[ A_y = r_A \times \sin(127^\circ) = 5.0 \times 0.7986 = 3.993 \approx 4.00\, \text{cm} \] --- ## Vector B - **Magnitude:** \( r_B = 3.5\, \text{cm} \) - **Angle:** \( \theta_B = 70^\circ \) ### Calculate \( \cos(70^\circ) \): \[ \cos(70^\circ) \approx 0.3420 \] ### Calculate \( \sin(70^\circ) \): \[ \sin(70^\circ) \approx 0.9397 \] ### Components: \[ B_x = 3.5 \times 0.3420 = 1.197 \approx 1.20\, \text{cm} \] \[ B_y = 3.5 \times 0.9397 = 3.289 \approx 3.29\, \text{cm} \] --- ## Vector C - **Magnitude:** \( r_C = 7.0\, \text{cm} \) - **Angle:** \( \theta_C = 20^\circ \) ### Calculate \( \cos(20^\circ) \): \[ \cos(20^\circ) \approx 0.9397 \] ### Calculate \( \sin(20^\circ) \): \[ \sin(20^\circ) \approx 0.3420 \] ### Components: \[ C_x = 7.0 \times 0.9397 = 6.578 \approx 6.58\, \text{cm} \] \[ C_y = 7.0 \times 0.3420 = 2.394 \approx 2.39\, \text{cm} \] --- ### **Final component forms of the vectors:** \[ \boxed{ \begin{aligned} \vec{A} &= -3.01\, \hat{x} + 4.00\, \hat{y} \\ \vec{B} &= 1.20\, \hat{x} + 3.29\, \hat{y} \\ \vec{C} &= 6.58\, \hat{x} + 2.39\, \hat{y} \end{aligned} } \] --- ### **Summary:** - **Vector A:** \(-3.01\, \hat{x} + 4.00\, \hat{y}\) - **Vector B:** \(1.20\, \hat{x} + 3.29\, \hat{y}\) - **Vector C:** \(6.58\, \hat{x} + 2.39\, \hat{y}\) Let me know if you'd like these in decimal form, with magnitude and angle, or further calculations!