Cross product Transitional Reasoning Download PDF

Questions

1. 1.

   In a physics laboratory experiment, a tracking system monitors an object of mass 2.0 kg moving in the xy-plane. At a specific moment, the object's position vector relative to the coordinate origin is $\vec{r} = (3.0\hat{i} + 4.0\hat{j})\text{ m}$ and its velocity vector is $\vec{v} = (5.0\hat{i} + 2.0\hat{j})\text{ m/s}$. The angular momentum vector with respect to the origin is defined by the cross product $\vec{L} = \vec{r} \times \vec{p}$, where $\vec{p} = m\vec{v}$ is the linear momentum. What is the angular momentum vector $\vec{L}$ of the object, and along which axis does it point?

   • A) $\vec{L} = +28\hat{k}\text{ kg}\cdot\text{m}^2/\text{s}$, pointing along the positive z-axis
   • B) $\vec{L} = -14\hat{k}\text{ kg}\cdot\text{m}^2/\text{s}$, pointing along the negative z-axis
   • C) $\vec{L} = -28\hat{k}\text{ kg}\cdot\text{m}^2/\text{s}$, pointing along the negative z-axis
   • D) $\vec{L} = +52\hat{k}\text{ kg}\cdot\text{m}^2/\text{s}$, pointing along the positive z-axis
2. 2.

   In an automated manufacturing system, a robotic probe pivots in the xy-plane about the origin with an instantaneous angular velocity of $\vec{\omega} = 4.0\hat{k}\text{ rad/s}$. The tip of the probe is currently located at position $\vec{r} = (0.60\hat{i} + 0.80\hat{j})\text{ m}$. The linear velocity vector of the probe tip is determined by the cross product relation $\vec{v} = \vec{\omega} \times \vec{r}$. Which of the following correctly gives the linear velocity vector $\vec{v}$ and the corresponding linear speed $|\vec{v}|$ of the probe tip?

   • A) $\vec{v} = (-3.2\hat{i} + 2.4\hat{j})\text{ m/s}$ and $|\vec{v}| = 4.0\text{ m/s}$
   • B) $\vec{v} = (3.2\hat{i} + 2.4\hat{j})\text{ m/s}$ and $|\vec{v}| = 4.0\text{ m/s}$
   • C) $\vec{v} = (2.4\hat{i} - 3.2\hat{j})\text{ m/s}$ and $|\vec{v}| = 4.0\text{ m/s}$
   • D) $\vec{v} = (-3.2\hat{i} - 2.4\hat{j})\text{ m/s}$ and $|\vec{v}| = 5.6\text{ m/s}$
3. 3.

   To model the solar energy collected by a planar sensor on an oceanographic buoy, researchers define two triangular boundary edges from a shared vertex using displacement vectors $\vec{a} = 6\hat{i}\text{ cm}$ and $\vec{b} = (2\hat{i} + 5\hat{j})\text{ cm}$. The geometric area of this triangular facet is given by $\frac{1}{2}|\vec{a} \times \vec{b}|$, while its normal orientation vector is defined by the cross product $\vec{a} \times \vec{b}$. Which of the following correctly provides the area of the sensor facet and the direction of its orientation vector?

   • A) $15\text{ cm}^2$, pointing along the $+\hat{k}$ direction
   • B) $30\text{ cm}^2$, pointing along the $+\hat{k}$ direction
   • C) $15\text{ cm}^2$, pointing along the $-\hat{k}$ direction
   • D) $30\text{ cm}^2$, pointing along the $-\hat{k}$ direction
4. 4.

   An aerospace engineering team models the rotational orientation of a satellite using two structural displacement vectors, $\vec{A} = (2\hat{i} + \hat{j} - 3\hat{k})\text{ m}$ and $\vec{B} = (-\hat{i} + 4\hat{k})\text{ m}$. To define a normal axis for an instrument panel, the team computes the vector cross product $\vec{C} = \vec{A} \times \vec{B}$. Which of the following correctly gives the resulting vector $\vec{C}$ and describes its geometric relationship to vectors $\vec{A}$ and $\vec{B}$?

   • A) $\vec{C} = (4\hat{i} + 11\hat{j} + \hat{k})\text{ m}^2$, and it lies within the plane formed by $\vec{A}$ and $\vec{B}$
   • B) $\vec{C} = (-4\hat{i} + 5\hat{j} - \hat{k})\text{ m}^2$, and it is perpendicular only to $\vec{A}$
   • C) $\vec{C} = (4\hat{i} + 5\hat{j} + \hat{k})\text{ m}^2$, and it is parallel to $\vec{B}$
   • D) $\vec{C} = (4\hat{i} - 5\hat{j} + \hat{k})\text{ m}^2$, and it is perpendicular to both $\vec{A}$ and $\vec{B}$
5. 5.

   A flight controller calculates the corrective steering vector $\vec{\tau} = \vec{r} \times \vec{F}$ acting on a drone maneuvered in a horizontal plane. The displacement vector $\vec{r}$ has a magnitude of 5.0 m pointing along the positive x-axis, while the applied thrust vector $\vec{F}$ has a magnitude of 8.0 N in the xy-plane oriented at an angle of 150° measured counterclockwise from the positive x-axis. Which of the following correctly gives the magnitude and direction of $\vec{\tau}$, as well as the consequence of reversing the operation to $\vec{F} \times \vec{r}$?

   • A) Magnitude is 34.6 N·m in the +z-direction, and $\vec{F} \times \vec{r}$ produces an identical vector because vector multiplication is commutative.
   • B) Magnitude is 20 N·m in the +y-direction, and $\vec{F} \times \vec{r}$ produces $-20\hat{j}$ N·m because the resulting vector must stay in the same plane.
   • C) Magnitude is 20 N·m in the +z-direction, and $\vec{F} \times \vec{r}$ produces $-20\hat{k}$ N·m because the cross product is anti-commutative.
   • D) Magnitude is 20 N·m in the −z-direction, and $\vec{F} \times \vec{r}$ produces $+20\hat{k}$ N·m because curling fingers from $\vec{r}$ to $\vec{F}$ points downward.