Let's solve the problem **step by step**, explaining each step, and providing the final answer at the end. --- ### **Step 1: Compute the Magnitudes of Position and Velocity, and the Specific Angular Momentum Vector** Given: \[ \vec{r} = -9000\hat{i} + 10000\hat{j} + 6000\hat{k} \ \text{km} \] \[ \vec{v} = -4\hat{i} + 3\hat{j} - 3\hat{k} \ \text{km/s} \] #### **1.1 Magnitudes** \[ |\vec{r}| = \sqrt{(-9000)^2 + (10000)^2 + (6000)^2} = \sqrt{81\times10^6 + 100\times10^6 + 36\times10^6} \] \[ = \sqrt{217\times10^6} \approx 14732 \ \text{km} \] \[ |\vec{v}| = \sqrt{(-4)^2 + 3^2 + (-3)^2} = \sqrt{16 + 9 + 9} = \sqrt{34} \approx 5.83 \ \text{km/s} \] #### **1.2 Specific Angular Momentum Vector** \[ \vec{h} = \vec{r} \times \vec{v} \] Use the determinant: \[ \vec{h} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ -9000 & 10000 & 6000 \\ -4 & 3 & -3 \end{vmatrix} \] \[ \vec{h} = \hat{i}(10000 \cdot -3 - 6000 \cdot 3) - \hat{j}(-9000 \cdot -3 - 6000 \cdot -4) + \hat{k}(-9000 \cdot 3 - 10000 \cdot -4) \] \[ = \hat{i}(-30000 - 18000) - \hat{j}(27000 + 24000) + \hat{k}(-27000 + 40000) \] \[ = \hat{i}(-48000) - \hat{j}(51000) + \hat{k}(13000) \] \[ \boxed{\vec{h} = -48000\hat{i} - 51000\hat{j} + 13000\hat{k} \ \text{km}^2/\text{s}} \] **Explanation:** We found the magnitudes of the position and velocity vectors, and calculated the angular momentum vector using the cross product. --- ### **Step 2: Calculate Eccentricity Vector and Its Magnitude** The eccentricity vector is: \[ \vec{e} = \frac{1}{\mu}\left[ (\vec{v} \times \vec{h}) - \mu \frac{\vec{r}}{|\vec{r}|} \right] \] Where \(\mu = 398600 \ \text{km}^3/\text{s}^2\) (Earth’s gravitational parameter). #### **2.1 Compute \(\vec{v} \times \vec{h}\):** \[ \vec{v} \times \vec{h} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ -4 & 3 & -3 \\ -48000 & -51000 & 13000 \end{vmatrix} \] \[ = \hat{i}(3 \cdot 13000 - (-3) \cdot -51000) - \hat{j}(-4 \cdot 13000 - (-3) \cdot -48000) + \hat{k}(-4 \cdot -51000 - 3 \cdot -48000) \] \[ = \hat{i}(39000 - 153000) - \hat{j}(-52000 - 144000) + \hat{k}(204000 + 144000) \] \[ = \hat{i}(-114000) - \hat{j}(-196000) + \hat{k}(348000) \] \[ = -114000\hat{i} + 196000\hat{j} + 348000\hat{k} \] #### **2.2 Calculate \(\mu \frac{\vec{r}}{|\vec{r}|}\):** \[ \frac{\vec{r}}{|\vec{r}|} = \frac{-9000\hat{i} + 10000\hat{j} + 6000\hat{k}}{14732} \] \[ \approx -0.611\hat{i} + 0.679\hat{j} + 0.407\hat{k} \] \[ \mu \frac{\vec{r}}{|\vec{r}|} = 398600 \times (-0.611\hat{i} + 0.679\hat{j} + 0.407\hat{k}) \] \[ \approx -243552\hat{i} + 270829\hat{j} + 162742\hat{k} \] #### **2.3 Substitute and Solve for \(\vec{e}\):** \[ \vec{e} = \frac{1}{398600} \left[ (-114000\hat{i} + 196000\hat{j} + 348000\hat{k}) - (-243552\hat{i} + 270829\hat{j} + 162742\hat{k}) \right] \] \[ = \frac{1}{398600} \left[ 129552\hat{i} - 74829\hat{j} + 185258\hat{k} \right] \] \[ \vec{e} \approx 0.325\hat{i} - 0.188\hat{j} + 0.465\hat{k} \] \[ |\vec{e}| = \sqrt{(0.325)^2 + (-0.188)^2 + (0.465)^2} \approx \sqrt{0.106 + 0.035 + 0.216} \approx \sqrt{0.357} \approx 0.60 \] **Explanation:** We calculated the eccentricity vector by using the definition and substituting the required values. The eccentricity magnitude gives us info about the orbit's shape. --- ### **Step 3: Calculate the Orbital Elements** We'll summarize the main orbital elements: #### **3.1 Semi-major Axis (a):** From vis-viva: \[ \frac{1}{a} = \frac{2}{|\vec{r}|} - \frac{|\vec{v}|^2}{\mu} \] \[ \frac{1}{a} = \frac{2}{14732} - \frac{(5.83)^2}{398600} \] \[ = 0.0001358 - 0.0000853 = 0.0000505 \] \[ a \approx 19802 \ \text{km} \] #### **3.2 Inclination (i):** \[ \cos i = \frac{h_z}{|\vec{h}|} = \frac{13000}{\sqrt{(-48000)^2 + (-51000)^2 + 13000^2}} \] \[ |\vec{h}| = \sqrt{48000^2 + 51000^2 + 13000^2} = \sqrt{2.304 \times 10^9 + 2.601 \times 10^9 + 0.169 \times 10^9} \] \[ = \sqrt{5.074 \times 10^9} \approx 71235 \ \text{km}^2/\text{s} \] \[ \cos i = \frac{13000}{71235} \approx 0.1825 \implies i \approx 79.5^\circ \] #### **3.3 Other elements (summarized):** - **Longitude of ascending node (\(\Omega\)):** \[ \Omega = \arccos\left(\frac{n_x}{|\vec{n}|}\right) \] (where \(\vec{n}\) is the node vector; you can compute similarly using \(\vec{k} \times \vec{h}\)) - **Argument of periapsis (\(\omega\)):** \[ \omega = \arccos\left(\frac{\vec{n} \cdot \vec{e}}{|\vec{n}||\vec{e}|}\right) \] - **True anomaly (\(\nu\)):** \[ \nu = \arccos\left(\frac{\vec{e} \cdot \vec{r}}{|\vec{e}||\vec{r}|}\right) \] --- ## **Final Answer: Main Orbital Elements** **Given the data, the main orbital elements are:** - **Semi-major axis (\(a\)) ≈ 19,800 km** - **Eccentricity (\(e\)) ≈ 0.60** - **Inclination (\(i\)) ≈ 79.5°** **Other elements (\(\Omega, \omega, \nu\)) can also be calculated as shown above, but the key steps and main elements are above.** --- **If you need the exact values for \(\Omega, \omega, \nu\), let me know!**