Cross product Transitional Reasoning Practice

Questions

1. 1.

   A navigation system on an exploration rover tracks two displacement vectors, $\vec{A}$ and $\vec{B}$, lying in the horizontal xy-plane. Vector $\vec{A}$ has a magnitude of 8.0 m directed along the +x-axis, while vector $\vec{B}$ has a magnitude of 5.0 m oriented at an angle of 30° counterclockwise from the +x-axis. To establish a vertical reference axis, the computer calculates the cross product $\vec{C} = \vec{A} \times \vec{B}$. What are the magnitude and direction of the resulting vector $\vec{C}$?

   • A) Magnitude of 35 m², pointing along the +z-axis
   • B) Magnitude of 20 m², pointing along the +z-axis
   • C) Magnitude of 20 m², pointing along the -z-axis
   • D) Magnitude of 40 m², pointing along the +z-axis
2. 2.

   During a wind turbine blade calibration, the relative orientation of two structural vectors is modeled by $\vec{A} = (2.0\hat{i} - 1.0\hat{j})\text{ m}$ and $\vec{B} = (1.0\hat{i} + 3.0\hat{j} - 2.0\hat{k})\text{ m}$. To establish an axis perpendicular to both structural elements, a technician calculates the vector cross product $\vec{C} = \vec{A} \times \vec{B}$. Which expression represents the resulting vector $\vec{C}$?

   • A) $\vec{C} = (-2.0\hat{i} - 4.0\hat{j} - 7.0\hat{k})\text{ m}^2$
   • B) $\vec{C} = (2.0\hat{i} - 4.0\hat{j} + 5.0\hat{k})\text{ m}^2$
   • C) $\vec{C} = (2.0\hat{i} - 4.0\hat{j} + 7.0\hat{k})\text{ m}^2$
   • D) $\vec{C} = (2.0\hat{i} + 4.0\hat{j} + 7.0\hat{k})\text{ m}^2$
3. 3.

   A technician configures an automated robotic arm that applies a torque defined by $\vec{\tau} = \vec{r} \times \vec{F}$ about a pivot at the origin. The position vector where the force acts is $\vec{r} = (2.0\hat{i})\text{ m}$, and the applied force vector is $\vec{F} = (4.0\hat{j} - 3.0\hat{k})\text{ N}$. Based on the properties of the vector cross product, which expression represents the resulting torque vector and its magnitude?

   • A) $\vec{\tau} = (6.0\hat{j} + 8.0\hat{k})\text{ N}\cdot\text{m}$ with a magnitude of 10 N·m
   • B) $\vec{\tau} = (-6.0\hat{j} + 8.0\hat{k})\text{ N}\cdot\text{m}$ with a magnitude of 10 N·m
   • C) $\vec{\tau} = (8.0\hat{j} - 6.0\hat{k})\text{ N}\cdot\text{m}$ with a magnitude of 10 N·m
   • D) $\vec{\tau} = (6.0\hat{j} - 8.0\hat{k})\text{ N}\cdot\text{m}$ with a magnitude of 10 N·m

Answers and step-by-step solutions appear after you finish the test.