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Vectors practice

QCE Specialist Maths · Original practice questions with worked solutions

Practise vector geometry, lines, planes and motion. Distinguish position vectors from direction vectors, and choose a scalar or vector equation that represents the condition in the question.

Key ideas

  • The scalar product is a⋅b=∣a∣∣b∣cos⁡θ\mathbf a\cdot\mathbf b=|\mathbf a||\mathbf b|\cos\theta. Non-zero perpendicular vectors have scalar product zero.
  • A line can be written r=a+td\mathbf r=\mathbf a+t\mathbf d. A plane with normal n\mathbf n through a\mathbf a satisfies n⋅(r−a)=0\mathbf n\cdot(\mathbf r-\mathbf a)=0.
  • For motion, differentiate position to get velocity and differentiate velocity to get acceleration. Speed is the magnitude of velocity.

Worked example

For a=(1,2,0)\mathbf a=(1,2,0) and b=(2,−1,3)\mathbf b=(2,-1,3), a⋅b=2−2+0=0\mathbf a\cdot\mathbf b=2-2+0=0. Both are non-zero, so they are perpendicular.

A common mistake

The angle between a line and a plane is complementary to the acute angle between the line’s direction and the plane’s normal.

Try these questions

Attempt each question before revealing the worked solution. Saved questions and marks also appear in the main bank on this device.

Q3 · Practice questionSimple familiar1 mark

QUESTION 3

A line, ll, and a plane, π\pi, are given by

l: r=(1−20)+λ(21−1),π: x−y+z=0.l:\ \mathbf r=\begin{pmatrix}1\\-2\\0\end{pmatrix}+\lambda\begin{pmatrix}2\\1\\-1\end{pmatrix},\qquad \pi:\ x-y+z=0.

The line

(A)
is parallel to, but does not lie in, the plane.
(B)
intersects the plane at one point.
(C)
lies entirely in the plane.
(D)
is perpendicular to the plane.
Question linkSyllabus coverage
Q8 · Practice questionComplex familiar5 marks

QUESTION 8 (5 marks)

A line, ll, and a plane, π\pi, are given by

l: r=(102)+λ(12−1),π: 2x−y+z=5.l:\ \mathbf r=\begin{pmatrix}1\\0\\2\end{pmatrix}+\lambda\begin{pmatrix}1\\2\\-1\end{pmatrix},\qquad \pi:\ 2x-y+z=5.
a)

Determine the point at which the line intersects the plane.

[2 marks]
b)

Determine the acute angle between the line and the plane, giving your answer in exact form.

[3 marks]
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Q9 · Practice questionComplex unfamiliar6 marks

QUESTION 9 (6 marks)

A particle moves in a plane with acceleration

a(t)=−2i^+6tj^,t≥0.\mathbf a(t)=-2\hat{\mathbf i}+6t\hat{\mathbf j},\qquad t\geq0.

Initially, v(0)=4i^−3j^\mathbf v(0)=4\hat{\mathbf i}-3\hat{\mathbf j} and r(0)=i^+2j^\mathbf r(0)=\hat{\mathbf i}+2\hat{\mathbf j}.

a)

Determine the position vector r(t)\mathbf r(t).

[4 marks]
b)

Show that at t=1t=1 the particle crosses the xx-axis with horizontal velocity.

[2 marks]
Question linkSyllabus coverage

All vectors practice questions

56 original questions · Page 1 of 3

  1. Q3 · Original practice · 1 mark
    Original Specialist Mathematics practice question 3: line and plane
    Vectors in three dimensions
  2. Q8 · Original practice · 5 marks
    Original Specialist Mathematics practice question 8: line-plane geometry
    Vectors in three dimensions
  3. Q9 · Original practice · 6 marks
    Original Specialist Mathematics practice question 9: motion in a plane
    Vector calculus
  4. Q24 · Original practice · 6 marks
    Let a=(4,−2,4)\mathbf a=(4,-2,4) and b=(1,2,2)\mathbf b=(1,2,2).
    Vectors in three dimensions
  5. Q26 · Original practice · 6 marks
    The sphere x2+y2+z2=9x^2+y^2+z^2=9 is intersected by the line r=(0,0,4)+t(1,0,−1).\mathbf r=(0,0,4)+t(1,0,-1).
    Vectors in three dimensions
  6. Q27 · Original practice · 1 mark
    A unit vector in the direction of a=(2,−2,1)\mathbf a=(2,-2,1) is
    Vectors in three dimensions
  7. Q28 · Original practice · 5 marks
    Points A(1,0,2)A(1,0,2), B(3,1,0)B(3,1,0) and C(0,2,1)C(0,2,1) lie in a plane.
    Vectors in three dimensions
  8. Q29 · Original practice · 1 mark
    Let a=(2,1,2)\mathbf a=(2,1,2) and b=(1,2,2)\mathbf b=(1,2,2). The scalar projection of a\mathbf a on b\mathbf b is
    Vectors in three dimensions
  9. Q30 · Original practice · 6 marks
    A tetrahedron has vertices with position vectors a,b,c,d\mathbf a,\mathbf b,\mathbf c,\mathbf d. Let MM and NN be the midpoints of ABAB and CDCD, respectively. Prove that the midpoint of MNMN has position vector a+b+c+d4,\frac{\mathbf a+\mathbf b+\mathbf c+\mathbf d}{4}, and hence show that the three line segments joining the midpoints of opposite edges of a tetrahe…
    Vectors in three dimensions
  10. Q31 · Original practice · 1 mark
    Point PP divides the segment from A(1,2,3)A(1,2,3) to B(7,−1,6)B(7,-1,6) internally in the ratio AP:PB=2:1AP:PB=2:1. The coordinates of PP are
    Vectors in three dimensions
  11. Q32 · Original practice · 7 marks
    Two particles move for t≥0t\ge0 with positions rA(t)=(2t,t),rB(t)=(6−t,4−2t).\mathbf r_A(t)=(2t,t),\qquad\mathbf r_B(t)=(6-t,4-2t).
    Vector calculus
  12. Q33 · Original practice · 1 mark
    The sphere x2+y2+z2−4x+6y−2z=11x^2+y^2+z^2-4x+6y-2z=11 has centre and radius
    Vectors in three dimensions
  13. Q34 · Original practice · 6 marks
    A particle follows the path r(t)=(3cos⁡t,2sin⁡t),0≤t≤2π.\mathbf r(t)=(3\cos t,2\sin t),\qquad0\le t\le2\pi.
    Vector calculus
  14. Q35 · Original practice · 1 mark
    The line r=(0,0,1)+t(1,2,−1)\mathbf r=(0,0,1)+t(1,2,-1) and the plane 2x−y=32x-y=3 are
    Vectors in three dimensions
  15. Q36 · Original practice · 6 marks
    A projectile is launched from level ground at 20 m s−120\ \mathrm{m\,s^{-1}} at 45∘45^\circ above the horizontal. Take g=10 m s−2g=10\ \mathrm{m\,s^{-2}} and ignore air resistance.
    Vector calculus
  16. Q37 · Original practice · 1 mark
    If a=(1,2,0)\mathbf a=(1,2,0) and b=(2,0,1)\mathbf b=(2,0,1) form adjacent sides of a parallelogram, its area is
    Vectors in three dimensions
  17. Q38 · Original practice · 5 marks
    A particle has position r(t)=5(cos⁡2t,sin⁡2t).\mathbf r(t)=5(\cos2t,\sin2t).
    Vector calculus
  18. Q39 · Original practice · 1 mark
    The parametric equations x=2+3tx=2+3t, y=−1+6ty=-1+6t describe the line
    Vector calculus
  19. Q41 · Original practice · 1 mark
    Particle AA follows rA(t)=(t,2t)\mathbf r_A(t)=(t,2t) and particle BB follows rB(t)=(3−t,1+t)\mathbf r_B(t)=(3-t,1+t). Which statement is correct?
    Vector calculus
  20. Q43 · Original practice · 1 mark
    For r(t)=(t2,et,sin⁡t)\mathbf r(t)=(t^2,e^t,\sin t), the acceleration at t=0t=0 is
    Vector calculus
  21. Q45 · Original practice · 1 mark
    A particle starts at the origin with velocity (1,3)(1,3) and constant acceleration (2,−1)(2,-1). Its position at t=2t=2 is
    Vector calculus
  22. Q64 · Original practice · 6 marks
    A sphere has centre C(2,−1,3)C(2,-1,3) and radius 5. It is cut by the plane 2x−y+2z=9.2x-y+2z=9. The intersection is a circle.
    Vector calculus
  23. Q66 · Original practice · 6 marks
    Two autonomous vehicles move in a plane, with position in kilometres after tt hours given by rA(t)=(2t,1+t),rB(t)=(12−t,7+0.5t),t≥0.\mathbf r_A(t)=(2t,1+t),\qquad\mathbf r_B(t)=(12-t,7+0.5t),\qquad t\ge0. Determine when they are closest together and find their minimum separation, giving answers to three decimal places where appropriate.
    Vector calculus
  24. Q68 · Original practice · 6 marks
    A projectile is launched from level ground with speed 30 m s−130\ \mathrm{m\,s^{-1}}. It must pass through the point (40,15)(40,15), where coordinates are in metres. Take g=10 m s−2g=10\ \mathrm{m\,s^{-2}} and ignore air resistance. Determine the two possible acute launch angles, to the nearest 0.1∘0.1^\circ.
    Vector calculus