Vector components Transitional Reasoning Download PDF

Questions

1. 1.

   A marine research vessel monitors subsurface ocean currents on a Cartesian grid where the positive x-axis points east and the positive y-axis points north. The acoustic sensor records a current velocity vector $\vec{v}$ with a magnitude of 25.0 cm/s flowing northwest, such that its northward component is $\frac{4}{3}$ times the magnitude of its westward component ($v_y = \frac{4}{3}|v_x|$). What are the scalar components $v_x$ and $v_y$ of this velocity vector, and what is its expression in Cartesian unit-vector notation?

   • A) $v_x = 15.0\text{ cm/s}$, $v_y = -20.0\text{ cm/s}$; $\vec{v} = (15.0\hat{i} - 20.0\hat{j})\text{ cm/s}$
   • B) $v_x = -20.0\text{ cm/s}$, $v_y = 15.0\text{ cm/s}$; $\vec{v} = (-20.0\hat{i} + 15.0\hat{j})\text{ cm/s}$
   • C) $v_x = -15.0\text{ cm/s}$, $v_y = -20.0\text{ cm/s}$; $\vec{v} = (-15.0\hat{i} - 20.0\hat{j})\text{ cm/s}$
   • D) $v_x = -15.0\text{ cm/s}$, $v_y = 20.0\text{ cm/s}$; $\vec{v} = (-15.0\hat{i} + 20.0\hat{j})\text{ cm/s}$
2. 2.

   A crate with a weight of 120 N rests on a ramp inclined at an angle of 30° above the horizontal. In physics, weight acts vertically downward and is resolved into two perpendicular components relative to the incline: one directed parallel to the ramp surface and one directed perpendicular to it. Given that $\sin 30^\circ = 0.50$ and $\cos 30^\circ = 0.87$, what are the magnitudes of the parallel component ($W_\parallel$) and the perpendicular component ($W_\perp$)?

   • A) $W_\parallel = 60\text{ N}$ and $W_\perp = 104.4\text{ N}$
   • B) $W_\parallel = 104.4\text{ N}$ and $W_\perp = 60\text{ N}$
   • C) $W_\parallel = 60\text{ N}$ and $W_\perp = 120\text{ N}$
   • D) $W_\parallel = 84.9\text{ N}$ and $W_\perp = 84.9\text{ N}$
3. 3.

   A rescue helicopter flies along a straight tracking path oriented $53^\circ$ north of east. A radar monitoring station records that the eastward component of the helicopter's displacement is 18.0 km. Given that $\sin 53^\circ = 0.80$ and $\cos 53^\circ = 0.60$, what are the northward displacement component ($d_y$) and the total straight-line distance ($d$) traveled by the helicopter?

   • A) $d_y$ = 13.5 km, $d$ = 22.5 km
   • B) $d_y$ = 14.4 km, $d$ = 18.0 km
   • C) $d_y$ = 24.0 km, $d$ = 42.0 km
   • D) $d_y$ = 24.0 km, $d$ = 30.0 km
4. 4.

   An oceanographic sensor station monitors coastal currents on a Cartesian grid where the positive x-axis points east and the positive y-axis points north. The station records a localized current velocity with scalar components $v_x = -3.0\text{ m/s}$ and $v_y = 3.0\sqrt{3}\text{ m/s}$. What are the magnitude of this velocity vector and its standard direction angle measured counterclockwise from the positive x-axis?

   • A) 6.0 m/s and 120°
   • B) 6.0 m/s and 60°
   • C) 8.2 m/s and 120°
   • D) 8.2 m/s and 240°
5. 5.

   An automated transport cart in a distribution center moves along a straight path from an initial position $P_1(16\text{ m}, 3\text{ m})$ to a final position $P_2(4\text{ m}, 12\text{ m})$ on a warehouse grid. What are the scalar components $\Delta x$ and $\Delta y$ of the cart's displacement vector, and what is the magnitude of this displacement?

   • A) $\Delta x = -12\text{ m}$, $\Delta y = +9\text{ m}$, and magnitude $= 15\text{ m}$
   • B) $\Delta x = +12\text{ m}$, $\Delta y = -9\text{ m}$, and magnitude $= 15\text{ m}$
   • C) $\Delta x = +20\text{ m}$, $\Delta y = +15\text{ m}$, and magnitude $= 25\text{ m}$
   • D) $\Delta x = -12\text{ m}$, $\Delta y = +9\text{ m}$, and magnitude $= 21\text{ m}$
6. 6.

   An engineering team is testing the stability of an anchor ring subjected to two planar forces, $\vec{A}$ and $\vec{B}$. Force $\vec{A}$ has the known Cartesian components $A_x = 8.0\text{ N}$ and $A_y = -3.0\text{ N}$. To ensure safe alignment, the resultant force $\vec{R} = \vec{A} + \vec{B}$ must act entirely in the positive y-direction with a magnitude of $7.0\text{ N}$. What must be the scalar components $B_x$ and $B_y$ of force $\vec{B}$?

   • A) $B_x = 8.0\text{ N}$ and $B_y = 4.0\text{ N}$
   • B) $B_x = -8.0\text{ N}$ and $B_y = 4.0\text{ N}$
   • C) $B_x = -8.0\text{ N}$ and $B_y = 10.0\text{ N}$
   • D) $B_x = 8.0\text{ N}$ and $B_y = 10.0\text{ N}$
7. 7.

   An autonomous underwater vehicle detects an ocean current with a speed of 10 m/s directed at an angle of 60° below the negative x-axis in the third quadrant. To compensate for the drift, the vehicle navigation system must resolve this velocity vector into its Cartesian scalar components. Given that $\cos 60^\circ = 0.50$ and $\sin 60^\circ = 0.87$, what are the scalar components $v_x$ and $v_y$ of the current velocity?

   • A) $v_x = +5.0$ m/s, $v_y = +8.7$ m/s
   • B) $v_x = -8.7$ m/s, $v_y = -5.0$ m/s
   • C) $v_x = -5.0$ m/s, $v_y = -8.7$ m/s
   • D) $v_x = -5.0$ m/s, $v_y = +8.7$ m/s
8. 8.

   During a laboratory investigation, a student pulls a test cart across a flat floor using a strap that exerts a tension force of 50 N directed upward and forward. The strap forms an angle of 37° with the vertical axis. Given that sin 37° = 0.60 and cos 37° = 0.80, what are the magnitudes of the horizontal component ($F_x$) and vertical component ($F_y$) of this tension force?

   • A) $F_x$ = 30 N and $F_y$ = 40 N
   • B) $F_x$ = 40 N and $F_y$ = 30 N
   • C) $F_x$ = 50 N and $F_y$ = 30 N
   • D) $F_x$ = 40 N and $F_y$ = 50 N