1. **Stating the problem:** Given the displacement vector $\Delta \vec{r} = \langle 1.10, 1.05, -1.70 \rangle$ m over a time interval $\Delta t = 780$ s, and velocity vector $\vec{v} = \langle 8.46, 8.07, -13.07 \rangle \times 10^{-3}$ m/s, we want to analyze the velocity, speed, and direction. 2. **Formulas and rules:** - Velocity vector: $\vec{v} = \frac{\Delta \vec{r}}{\Delta t}$ - Speed: magnitude of velocity $v = |\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}$ - Unit vector of velocity: $\hat{v} = \frac{\vec{v}}{|\vec{v}|}$ - Angle with x-axis: $\theta_x = \cos^{-1}\left(\frac{v_x}{|\vec{v}|}\right)$ 3. **Check velocity from displacement and time:** $$\vec{v} = \frac{\langle 1.10, 1.05, -1.70 \rangle}{780} = \langle \frac{1.10}{780}, \frac{1.05}{780}, \frac{-1.70}{780} \rangle = \langle 0.00141, 0.00135, -0.00218 \rangle \text{ m/s}$$ 4. **Given velocity vector:** $$\vec{v} = \langle 8.46, 8.07, -13.07 \rangle \times 10^{-3} = \langle 0.00846, 0.00807, -0.01307 \rangle \text{ m/s}$$ 5. **Calculate speed from given velocity:** $$v = \sqrt{(0.00846)^2 + (0.00807)^2 + (-0.01307)^2} = \sqrt{7.16 \times 10^{-5} + 6.51 \times 10^{-5} + 1.71 \times 10^{-4}} = \sqrt{3.07 \times 10^{-4}} = 0.0175 \text{ m/s}$$ 6. **Given speed is $2.92 \times 10^{-3}$ m/s, which differs from calculated speed $0.0175$ m/s, so verify data or units.** 7. **Calculate unit vector of velocity:** $$\hat{v} = \frac{1}{0.0175} \langle 0.00846, 0.00807, -0.01307 \rangle = \langle 0.484, 0.461, -0.747 \rangle$$ 8. **Given unit vector is $\langle 3.71, 3.53, -5.73 \rangle$ m/s, which is not a unit vector (magnitude > 1), so likely incorrect or misinterpreted.** 9. **Calculate angle with x-axis:** $$\theta_x = \cos^{-1}\left(\frac{v_x}{|\vec{v}|}\right) = \cos^{-1}\left(\frac{0.00846}{0.0175}\right) = \cos^{-1}(0.484) = 61.1^\circ$$ **Final answers:** - Velocity from displacement/time: $\langle 0.00141, 0.00135, -0.00218 \rangle$ m/s - Speed from given velocity: $0.0175$ m/s - Unit vector of velocity: $\langle 0.484, 0.461, -0.747 \rangle$ - Angle with x-axis: $61.1^\circ$ Note: Some given values (speed, unit vector) do not match calculations, suggesting possible data inconsistencies.