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Vectors practice

QCE Specialist Maths · Original practice questions with worked solutions

All vectors practice questions

56 original questions · Page 2 of 3

  1. Q105 · Original practice · 1 mark
    A projectile is launched from level ground at 25 m s−125\ \mathrm{m\,s^{-1}} at 40∘40^\circ above the horizontal. Taking g=9.8 m s−2g=9.8\ \mathrm{m\,s^{-2}}, its horizontal range is approximately
    Vector calculus
  2. Q107 · Original practice · 1 mark
    A particle has r(t)=(4cos⁡2t,4sin⁡2t)\mathbf r(t)=(4\cos2t,4\sin2t). The magnitude of its acceleration is
    Vector calculus
  3. Q108 · Original practice · 7 marks
    A drone and a moving beacon have positions rD(t)=(2t,t),rB(t)=(8cos⁡(0.4t),8sin⁡(0.4t)),0≤t≤6.\begin{aligned}\mathbf r_D(t)&=(2t,t),\\\mathbf r_B(t)&=(8\cos(0.4t),8\sin(0.4t)),\end{aligned}\qquad0\le t\le6. Coordinates are in kilometres and tt is in hours.
    Vectors in three dimensions
  4. Q109 · Original practice · 1 mark
    For r(t)=(3cos⁡t,2sin⁡t)\mathbf r(t)=(3\cos t,2\sin t), the angle between velocity and acceleration at t=π/4t=\pi/4 is approximately
    Vector calculus
  5. Q110 · Original practice · 6 marks
    For a real parameter tt, define A=(t,0,1),B=(1,t,0),C=(0,1,t).A=(t,0,1),\qquad B=(1,t,0),\qquad C=(0,1,t).
    Vectors in three dimensions
  6. Q113 · Original practice · 1 mark
    A direction vector from A=(2,−1,0)A=(2,-1,0) to B=(5,1,−4)B=(5,1,-4) is
    Vectors in three dimensions
  7. Q116 · Original practice · 1 mark
    The area of the parallelogram generated by a=(1,2,0)a=(1,2,0) and b=(0,1,2)b=(0,1,2) is
    Vectors in three dimensions
  8. Q120 · Original practice · 1 mark
    Two non-parallel lines in 3D3D intersect if
    Vectors in three dimensions
  9. Q122 · Original practice · 6 marks
    The lines l1l_{1}: r=(1,2,0)+s(1,−1,2)r=(1,2,0)+s(1,-1,2) and l2l_{2}: r=(3,0,1)+t(2,1,−1)r=(3,0,1)+t(2,1,-1) are given.
    Vectors in three dimensions
  10. Q133 · Original practice · 1 mark
    For a=(1,2,2),b=(2,0,1)a=(1,2,2), b=(2,0,1), the scalar projection of aa onto bb is
    Vectors in three dimensions
  11. Q138 · Original practice · 1 mark
    For a=(2,1,0),b=(1,−1,2),∣a×b∣a=(2,1,0), b=(1,-1,2), |a \times b| is
    Vectors in three dimensions
  12. Q141 · Original practice · 6 marks
    A particle has position r(t)=(t2,2t,3−t2)r(t)=(t^{2}, 2t, 3-t^{2}) for t≥0t\ge 0.
    Vector calculus
  13. Q149 · Original practice · 6 marks
    Two particles move in the plane for 0≤t≤40\leq t\leq4, where tt is measured in seconds. Their position vectors, in metres, are
    rA(t)=ti+(t−2)2j,rB(t)=(t+1)i+j.\mathbf r_A(t)=t\mathbf i+(t-2)^2\mathbf j,\qquad \mathbf r_B(t)=(t+1)\mathbf i+\mathbf j.
    Their paths are shown. The points A0A_0 and B0B_0 indicate their positions at t=0t=0.
    Vector calculus
  14. Q155 · Original practice · 6 marks
    A point PP has coordinates (4,1,3)(4,1,3). A plane Π\Pi has equation
    x+2y+2z=3.x+2y+2z=3.
    The point HH is the perpendicular projection of PP onto Π\Pi. The point P′P' is the reflection of PP in Π\Pi, so HH is the midpoint of PP′PP'. The diagram is schematic.
    Vectors in two and three dimensions
  15. Q188 · Original practice · 5 marks
    A point P=(2,1,2)P=(2,1,2) and a line L:r=(1,0,0)+t(1,2,2)L:\mathbf r=(1,0,0)+t(1,2,2) are given, with t∈Rt\in\mathbb R.
    Vectors in two and three dimensions
  16. Q189 · Original practice · 5 marks
    A triangular panel has vertices A=(1,0,0)A=(1,0,0), B=(0,2,0)B=(0,2,0) and C=(0,0,3)C=(0,0,3).
    Vectors in two and three dimensions
  17. Q190 · Original practice · 8 marks
    A laser ray follows r=(1,0,2)+t(2,1,−1)\mathbf r=(1,0,2)+t(2,1,-1), t≥0t\ge0. A reflecting screen lies in x+y+z=6x+y+z=6.
    Vectors in two and three dimensions
  18. Q191 · Original practice · 6 marks
    Two straight rails in a sculpture are modelled by L1:(x,y,z)=(t,t,0)L_1:(x,y,z)=(t,t,0) and L2:(x,y,z)=(s,1−s,1)L_2:(x,y,z)=(s,1-s,1), with s,t∈Rs,t\in\mathbb R.
    Vectors in two and three dimensions
  19. Q192 · Original practice · 5 marks
    A river is 100 m wide, measured due north. Its current is 3i3\mathbf i m/s east. A boat has speed 5 m/s relative to the water. Let j\mathbf j point north. Ignore acceleration and current changes.
    Vectors in two and three dimensions
  20. Q193 · Original practice · 6 marks
    Two animated drones have positions rA=(t,2t)\mathbf r_A=(t,2t) and rB=(6−t,3)\mathbf r_B=(6-t,3) metres for 0≤t≤30\le t\le3 seconds. Their paths are shown.
    Vector calculus
  21. Q194 · Original practice · 5 marks
    For 0≤t≤30\le t\le3, a particle has position r(t)=(t+1)i+(t+1)−1j\mathbf r(t)=(t+1)\mathbf i+(t+1)^{-1}\mathbf j metres.
    Vector calculus
  22. Q195 · Original practice · 1 mark
    Particles move on the radius-two circle with rA=2(cos⁡t,sin⁡t)\mathbf r_A=2(\cos t,\sin t) and rB=2(cos⁡(2t+π),sin⁡(2t+π))\mathbf r_B=2(\cos(2t+\pi),\sin(2t+\pi)). What is their first meeting time after t=0?
    Vector calculus
  23. Q196 · Original practice · 10 marks
    A camera rig has velocity v(t)=2ti+3t2j\mathbf v(t)=2t\mathbf i+3t^2\mathbf j m/s on 0≤t≤10\le t\le1, and initial position (1,-1) m.
    Vector calculus
  24. Q202 · Original practice · 1 mark
    Which plane contains all points equidistant from A=(1,0,2)A=(1,0,2) and B=(3,2,0)B=(3,2,0)?
    Vectors in two and three dimensions