Dot product Transitional Reasoning Download PDF

Questions

1. 1.

   A structural engineering module analyzes the load distribution at a truss joint subjected to two concurrent force vectors, $\vec{A}$ and $\vec{B}$. The vectors have magnitudes $|\vec{A}| = 6.0\text{ N}$ and $|\vec{B}| = 4.0\text{ N}$, with an angle of $60^\circ$ between their directions. To evaluate joint stability, the system calculates the combined scalar product $S = (\vec{A} + 2\vec{B}) \cdot (2\vec{A} - \vec{B})$. What is the value of this scalar product?

   • A) $40.0\text{ N}^2$
   • B) $-20.0\text{ N}^2$
   • C) $52.0\text{ N}^2$
   • D) $76.0\text{ N}^2$
2. 2.

   An autonomous surface vehicle navigates near a harbor entrance where its velocity vector is $\vec{v} = (5.0\hat{i} + 10.0\hat{j})\text{ m/s}$. Control software defines the shipping channel orientation using the reference vector $\vec{u} = (4.0\hat{i} + 3.0\hat{j})\text{ m}$. To correct for cross-channel drift, the guidance computer calculates the perpendicular velocity vector $\vec{v}_{\perp} = \vec{v} - \vec{v}_{\parallel}$, where $\vec{v}_{\parallel}$ is the vector projection of $\vec{v}$ onto $\vec{u}$. What is the perpendicular velocity component vector $\vec{v}_{\perp}$?

   • A) $(-3.0\hat{i} + 4.0\hat{j})\text{ m/s}$
   • B) $(8.0\hat{i} + 6.0\hat{j})\text{ m/s}$
   • C) $(3.0\hat{i} - 4.0\hat{j})\text{ m/s}$
   • D) $(1.0\hat{i} + 7.0\hat{j})\text{ m/s}$
3. 3.

   During a flight-control calibration test, a drone's onboard computer records its forward velocity vector $\vec{v}$ and an ambient wind velocity vector $\vec{w}$. Sensor logs reveal that the resultant velocity magnitude when steering with the crosswind equals the resultant magnitude when steering against it, such that $|\vec{v} + \vec{w}| = |\vec{v} - \vec{w}|$. Using the algebraic properties of the dot product, what geometric condition must non-zero vectors $\vec{v}$ and $\vec{w}$ satisfy?

   • A) The vectors must have identical magnitudes, $|\vec{v}| = |\vec{w}|$.
   • B) The vectors must be perpendicular to each other, so $\vec{v} \cdot \vec{w} = 0$.
   • C) The vectors must be parallel and point in the same direction, so $\vec{v} \cdot \vec{w} = |\vec{v}||\vec{w}|$.
   • D) The vectors must be anti-parallel and point in opposite directions, so $\vec{v} \cdot \vec{w} = -|\vec{v}||\vec{w}|$.
4. 4.

   A monitoring system tracks two concurrent forces acting on a structural node: $\vec{F}_1$ with a magnitude of 5.0 N and $\vec{F}_2$ with a magnitude of 8.0 N. Diagnostic telemetry reports that the scalar product of the two force vectors is $\vec{F}_1 \cdot \vec{F}_2 = -20.0\text{ N}^2$. What is the magnitude of the resultant force $\vec{F}_{\text{net}} = \vec{F}_1 + \vec{F}_2$ acting on the node?

   • A) 7.0 N
   • B) 9.4 N
   • C) 11.4 N
   • D) 13.0 N
5. 5.

   An autonomous underwater survey vehicle uses two internal orientation vectors to establish its reference grid: a forward track vector $\vec{u} = (4.0\hat{i} - 6.0\hat{j})\text{ m}$ and a lateral probe vector $\vec{w} = (k\hat{i} + 8.0\hat{j})\text{ m}$. To ensure that the coordinate grid is properly calibrated, the lateral probe must be strictly perpendicular to the forward track. What value of the scalar parameter $k$ satisfies the condition that $\vec{u}$ and $\vec{w}$ are orthogonal?

   • A) $k = -12.0$
   • B) $k = 12.0$
   • C) $k = -5.3$
   • D) $k = 3.0$
6. 6.

   A robotic positioning system moves an end-effector along a linear track defined by the displacement vector $\vec{r} = (3.0\hat{i} + 4.0\hat{j})\text{ cm}$. An actuator applies a force $\vec{F} = (4.0\hat{i} + 7.0\hat{j})\text{ N}$ to guide the tool along this path. To evaluate the effective thrust along the track, the control system determines the scalar projection of $\vec{F}$ onto $\vec{r}$. What is the magnitude of this scalar component along the direction of the track?

   • A) 40.0 N
   • B) 11.0 N
   • C) 1.6 N
   • D) 8.0 N
7. 7.

   In an automated distribution center, two mobile robots travel along straight tracks represented by displacement vectors $\vec{d}_1 = (2.0\hat{i} + 1.0\hat{j})\text{ m}$ and $\vec{d}_2 = (1.0\hat{i} + 3.0\hat{j})\text{ m}$. A tracking system calculates the scalar product $\vec{d}_1 \cdot \vec{d}_2$ to determine the geometric relationship between their trajectories. What are the value of the scalar product $\vec{d}_1 \cdot \vec{d}_2$ and the angle $\theta$ between the paths of the two robots?

   • A) $5.0\text{ m}^2$ and $45^\circ$
   • B) $7.0\text{ m}^2$ and $30^\circ$
   • C) $5.0\text{ m}^2$ and $60^\circ$
   • D) $1.0\text{ m}^2$ and $75^\circ$
8. 8.

   A sensor package on an autonomous research vehicle experiences a constant force $\vec{F} = (5.0\hat{i} - 2.0\hat{j} + 1.0\hat{k})$ N during a straight-line displacement $\vec{d} = (2.0\hat{i} + 3.0\hat{j} + 4.0\hat{k})$ m. A navigation system calculates the scalar product $\vec{F} \cdot \vec{d}$ to monitor the mechanical work transferred to the package. What is the value of $\vec{F} \cdot \vec{d}$, and what does its sign indicate about the angle $\theta$ between the force and displacement vectors?

   • A) +8.0 N·m; the angle $\theta$ between the force and displacement is obtuse ($\theta > 90^\circ$).
   • B) +8.0 N·m; the angle $\theta$ between the force and displacement is acute ($\theta < 90^\circ$).
   • C) +20.0 N·m; the angle $\theta$ between the force and displacement is acute ($\theta < 90^\circ$).
   • D) −4.0 N·m; the angle $\theta$ between the force and displacement is obtuse ($\theta > 90^\circ$).