Vector addition and subtraction Transitional Reasoning Download PDF

Questions

1. 1.

   An autonomous harbor vessel utilizes two horizontal thrusters to maintain station during docking maneuvers. Thruster 1 delivers a force $\vec{F}_1$ of magnitude 600 N oriented along the positive x-axis. Thruster 2 provides a force $\vec{F}_2$ of magnitude 1000 N. If the resultant force vector $\vec{F}_{\text{net}} = \vec{F}_1 + \vec{F}_2$ acts entirely along the positive y-axis, what is the magnitude of $\vec{F}_{\text{net}}$?

   • A) 400 N
   • B) 800 N
   • C) 1166 N
   • D) 1600 N
2. 2.

   A materials testing laboratory applies two coplanar tension forces, $\vec{F}_1$ and $\vec{F}_2$, to a central anchor ring. By varying the angle between the two forces, the engineers determine that the maximum possible magnitude of their resultant is 17 N, whereas the minimum possible magnitude of their resultant is 7 N. If the apparatus is adjusted so that the two forces act perpendicular to each other, what is the magnitude of their resultant force?

   • A) 10 N
   • B) 12 N
   • C) 13 N
   • D) 15 N
3. 3.

   An engineering test rig evaluates a dual-cable support bracket subjected to two coplanar tension force vectors, $\vec{T}_1$ and $\vec{T}_2$, each having an identical magnitude of 15 N. A digital load cell records that the magnitude of their vector sum, $|\vec{T}_1 + \vec{T}_2|$, is 24 N. To evaluate transverse shear stress across the mounting pin, the analysis software must determine the magnitude of the vector difference, $|\vec{T}_1 - \vec{T}_2|$. What is the magnitude of this vector difference?

   • A) 9 N
   • B) 18 N
   • C) 12 N
   • D) 30 N
4. 4.

   An environmental monitoring buoy in a coastal bay is subjected to two steady horizontal forces, $\vec{F}_1$ and $\vec{F}_2$, caused by surface wind and tidal currents. Field sensors record that the magnitude of the vector sum of these forces is identical to the magnitude of their vector difference, such that $|\vec{F}_1 + \vec{F}_2| = |\vec{F}_1 - \vec{F}_2|$. Given that the individual force magnitudes are measured as $|\vec{F}_1| = 6.0\text{ N}$ and $|\vec{F}_2| = 8.0\text{ N}$, what is the magnitude of the net horizontal force vector $\vec{F}_{\text{net}} = \vec{F}_1 + \vec{F}_2$ acting on the buoy?

   • A) 2.0 N
   • B) 7.0 N
   • C) 14.0 N
   • D) 10.0 N
5. 5.

   A robotic positioning arm operates in a two-dimensional horizontal workspace. The position vector of its primary joint relative to the base origin is $\vec{r}_1$, having a magnitude of 10 cm along the positive x-axis. The position vector of its secondary joint relative to the base origin is $\vec{r}_2$, which also has a magnitude of 10 cm but is oriented at $60^\circ$ counterclockwise from the positive x-axis. What are the magnitude and direction of the relative displacement vector $\Delta\vec{r} = \vec{r}_2 - \vec{r}_1$ pointing from the primary joint to the secondary joint?

   • A) Magnitude of 10 cm, directed at $60^\circ$ counterclockwise from the positive x-axis
   • B) Magnitude of $10\sqrt{3}$ cm, directed at $120^\circ$ counterclockwise from the positive x-axis
   • C) Magnitude of 10 cm, directed at $120^\circ$ counterclockwise from the positive x-axis
   • D) Magnitude of $10\sqrt{3}$ cm, directed at $30^\circ$ counterclockwise from the positive x-axis
6. 6.

   During a flight test in a horizontal plane, a monitoring drone changes its velocity from an initial vector of $\vec{v}_i = (12\hat{i} - 5\hat{j})\text{ m/s}$ to a final vector of $\vec{v}_f = (4\hat{i} + 10\hat{j})\text{ m/s}$. The onboard navigation system evaluates the change in velocity using vector subtraction, defined by $\Delta\vec{v} = \vec{v}_f - \vec{v}_i$. What is the magnitude of the velocity change vector $\Delta\vec{v}$?

   • A) 7 m/s
   • B) 17 m/s
   • C) 23 m/s
   • D) 13 m/s
7. 7.

   An automated underwater rover navigates a horizontal seabed by executing two consecutive straight-line displacements, $\vec{d}_1$ and $\vec{d}_2$. In the first leg of its journey, the rover records a displacement of $\vec{d}_1 = (5\hat{i} - \hat{j})\text{ m}$. To reach its target sampling station, the rover's total displacement from the origin must be $\vec{d}_{\text{net}} = (\hat{i} + 2\hat{j})\text{ m}$. Which displacement vector $\vec{d}_2$ must the control system command for the second leg?

   • A) $\vec{d}_2 = (6\hat{i} + \hat{j})\text{ m}$
   • B) $\vec{d}_2 = (4\hat{i} - 3\hat{j})\text{ m}$
   • C) $\vec{d}_2 = (-4\hat{i} + 3\hat{j})\text{ m}$
   • D) $\vec{d}_2 = (-4\hat{i} - 3\hat{j})\text{ m}$