Vectors · CAPE Pure Mathematics Unit 1

94 past-paper questions on Vectors, part of Trigonometry, Geometry and Vectors, from every CAPE Pure Mathematics Unit 1 paper on Quelpr.

1. Q161 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1If \mathbf{a} = 5\mathbf{i} + \mathbf{j} and \mathbf{b} = \lambda\mathbf{i} + 5\mathbf{j} are parallel vectors, then the value of \lambda is
2. Q261 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (Rest of Region) · Paper 1The point A has coordinates (3, -2). The vector 3\vec{OA} is
3. Q281 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1Given that the vector (k+1)\mathbf{i} + 3\mathbf{j} is parallel to the vector 2\mathbf{i} - 6\mathbf{j}, the value of k is
4. Q291 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1Given that \mathbf{p} = \begin{pmatrix} -2 \\ 3 \end{pmatrix} and \mathbf{r} = \begin{pmatrix} 5 \\ -2 \end{pmatrix}, then \mathbf{p} \cdot \mathbf{r} equals
5. Q301 mark · multiple choice· CAPE Pure Mathematics Unit 1 · 2008 (T&T) · Paper 1The cosine of the angle between the vectors -6\mathbf{j} and \mathbf{i} + \mathbf{j} is
6. 4(b)(i)3 marks· CAPE Pure Mathematics Unit 1 · 2022 · Paper 2Determine the equation of the plane that passes through the point (0, 2, -2) and which is perpendicular to the vector \mathbf{v} = 3\mathbf{i} + 2\mathbf{j} + 4\mathbf{k}.
7. 4(b)(ii)3 marks· CAPE Pure Mathematics Unit 1 · 2022 · Paper 2Determine the angle between the vector 3\mathbf{i} - 2\mathbf{j} and the x-axis.
8. 4(c)5 marks· CAPE Pure Mathematics Unit 1 · 2022 · Paper 2The parametric equations of a line are given as x = 2 + 5\lambda, y = -3 + 4\lambda, z = 4 - 2\lambda. Determine the coordinates of the point where the line crosses the xy plane.
9. 9(a)3 marks· CAPE Pure Mathematics Unit 1 · May/June 2005 · Paper 1Express the position vector of EACH of A, B and C in terms of i and j.
10. 9(b)6 marks· CAPE Pure Mathematics Unit 1 · May/June 2005 · Paper 1If vec(AB) = vec(CD), find the position vector of D in terms of i and j.
11. 107 marks· CAPE Pure Mathematics Unit 1 · May/June 2005 · Paper 1Find the values of theta, 0 <= theta <= 2pi, for which the vectors cos(theta)i + sqrt(3)j and (1/4)i + sin(theta)j are parallel.
12. 10(a)7 marks· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Find x, y in R such that xp + yq = -3i - 11j.
13. 10(b)2 marks· CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 1Show that p and q are perpendicular.
14. Q201 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1The vector \begin{pmatrix} p \\ q \end{pmatrix} is perpendicular to the vector \begin{pmatrix} 3 \\ -1 \end{pmatrix}. The relation between p and q is
15. Q261 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1A vector equation is given as s\begin{pmatrix} -2 \\ 1 \end{pmatrix} + t\begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} -5 \\ 1 \end{pmatrix}. The values of s and t are, respectively
16. Q271 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1If p = 2\mathbf{i} + \mathbf{j} and q = \lambda\mathbf{i} + 6\mathbf{j} are perpendicular vectors, then the value of \lambda is
17. Q301 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2009 · Paper 1The vector \mathbf{u} has magnitude 4\sqrt{5} units and is parallel to the vector \mathbf{v} = \mathbf{i} - 2\mathbf{j}. A unit vector parallel to \mathbf{u} is
18. Q191 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1The point P has position vector \begin{pmatrix}-3 \\ 5\end{pmatrix} and Q is a point such that \vec{PQ} = \begin{pmatrix}1 \\ -7\end{pmatrix}. The position vector of Q is
19. Q201 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1The vector \mathbf{a} is given as 5\mathbf{i} + 12\mathbf{j}. A unit vector parallel to \mathbf{a} is
20. Q221 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1If x = r\sin\theta\cos\alpha, y = r\sin\theta\sin\alpha, z = r\cos\theta, then
21. Q261 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1Given that \mathbf{a} = \begin{pmatrix}-3 \\ 7\end{pmatrix} and \mathbf{b} = \begin{pmatrix}2 \\ -1\end{pmatrix}, then |3\mathbf{a} + 2\mathbf{b}| is equal to
22. Q281 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 1\vec{OP} and \vec{OQ} are two vectors such that \vec{OP} = \mathbf{r} - \mathbf{s} and \vec{OQ} = 2\mathbf{r} + 3\mathbf{s}. Given that \vec{OP} is perpendicular to \vec{OQ} then \mathbf{r}\cdot\mathbf{s}…
23. Q161 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1The cosine of the angle between the vectors -6\,\mathbf{j} and \mathbf{i} + \mathbf{j} is
24. Q261 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1A vector equation is given as s\begin{pmatrix} 1 \\ -2 \end{pmatrix} + t\begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} -5 \\ -1 \end{pmatrix}. The values of s and t are, respectively
25. Q271 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2011 · Paper 1If \mathbf{p} = 2\,\mathbf{i} + \mathbf{j} and \mathbf{q} = \lambda\,\mathbf{i} + 6\,\mathbf{j} are perpendicular vectors, then the value of \lambda is
26. Q181 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1The vector \begin{pmatrix} p \\ q \end{pmatrix} is perpendicular to the vector \begin{pmatrix} 3 \\ -1 \end{pmatrix}. The relationship between p and q is
27. Q191 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1The point P has position vector \begin{pmatrix} -3 \\ 5 \end{pmatrix} and Q is a point such that \vec{PQ} = \begin{pmatrix} 1 \\ -7 \end{pmatrix}. The position vector of Q is
28. Q241 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1A vector equation is given as s\begin{pmatrix} -2 \\ 1 \end{pmatrix} + t\begin{pmatrix} -1 \\ 1 \end{pmatrix} = \begin{pmatrix} -5 \\ 1 \end{pmatrix}. The values of s and t are, respectively
29. Q281 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2012 · Paper 1The vector \mathbf{u} has magnitude 4\sqrt{5} units and is parallel to the vector \mathbf{v} = \mathbf{i} - 2\mathbf{j}. A unit vector parallel to \mathbf{u} is
30. Q241 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1With respect to an origin O, A has coordinates (3, -2). The position vector of 3\,\vec{OA} is
31. Q271 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1If \mathbf{p} = 2\mathbf{i} + \mathbf{j} and \mathbf{q} = \lambda\mathbf{i} + 6\mathbf{j} are perpendicular vectors, then the value of \lambda is
32. Q291 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2013 · Paper 1The cosine of the angle between the vectors -6\,\mathbf{j} and \mathbf{i} + \mathbf{j} is
33. Q181 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1The vector \begin{pmatrix} p \\ q \end{pmatrix} is perpendicular to the vector \begin{pmatrix} 3 \\ -1 \end{pmatrix}. The relationship between p and q is
34. Q271 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1Given that \mathbf{a} = \begin{pmatrix} -3 \\ 7 \end{pmatrix} and \mathbf{b} = \begin{pmatrix} 2 \\ -1 \end{pmatrix}, then |3\mathbf{a} + 2\mathbf{b}| is equal to
35. Q301 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 1The vector \mathbf{a} is given as 5\mathbf{i} + 12\mathbf{j}. A unit vector parallel to \mathbf{a} is
36. Q211 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1With respect to an origin O, A has coordinates (3, -2). The position vector of 3\,\vec{OA} is
37. Q241 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1The vector \begin{pmatrix} p \\ q \end{pmatrix} is perpendicular to the vector \begin{pmatrix} 3 \\ -1 \end{pmatrix}. The relationship between p and q is
38. Q271 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1A vector equation is given as s\begin{pmatrix} -2 \\ 1 \end{pmatrix} + t\begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} -5 \\ -1 \end{pmatrix}. The values of s and t are, respectively
39. Q281 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1The point P has position vector \begin{pmatrix} -3 \\ 5 \end{pmatrix} and Q is a point such that \vec{PQ} = \begin{pmatrix} 1 \\ -7 \end{pmatrix}. The position vector of Q is
40. Q291 mark · multiple choice· CAPE Pure Mathematics Unit 1 · May/June 2015 · Paper 1The cosine of the angle between the vectors -6\mathbf{j} and \mathbf{i} + \mathbf{j} is
41. 4(b)6 marks· CAPE Pure Mathematics Unit 1 · May/June 2023 · Paper 2The equation of a line is x = 2 + t, y = 1 - 3t and z = 4 + t, and the equation of a plane is x + 2y + z = 12. Determine the point of intersection of the line and the plane.
42. 4(c)(i)4 marks· CAPE Pure Mathematics Unit 1 · May/June 2023 · Paper 2Determine the vector equation of the plane which passes through (1, 5, -1) and which is perpendicular to the vector \begin{pmatrix}2\\4\\3\end{pmatrix}.
43. 4(c)(ii)3 marks· CAPE Pure Mathematics Unit 1 · May/June 2023 · Paper 2Hence, determine the coordinates of the point in the plane where y = 3 and z = 1.
44. 4(d)7 marks· CAPE Pure Mathematics Unit 1 · May/June 2023 · Paper 2Given that a line is parallel to the vector u = \begin{pmatrix}1\\-3\\1\end{pmatrix} and that the vector v = \begin{pmatrix}1\\2\\1\end{pmatrix} is normal to the plane, calculate the angle between the line and the plane.
45. 4(a)(i)4 marks· CAPE Pure Mathematics Unit 1 · May/June 2024 · Paper 2Show that the equation of the plane is 3x - 4y + 2z - 5 = 0.
46. 4(a)(ii)6 marks· CAPE Pure Mathematics Unit 1 · May/June 2024 · Paper 2Calculate the angle, in radians, between P and the plane with equation 7x - 4y + 3z = 5.
47. 4(b)(i)2 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Calculate a · b.
48. 4(b)(ii)4 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Hence, calculate the angle between the vectors a and b.
49. 4(c)(i)2 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Determine the displacement vector AB in terms of i, j and k.
50. 4(c)(ii)2 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Determine a unit vector parallel to AB in terms of i, j and k.
51. 4(c)(iii)3 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Determine the Cartesian equation of the line passing through A and B.
52. 4(d)4 marks· CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2Determine the vector equation of the plane that passes through the point (1, -1, 2) and that is perpendicular to the vector 2i + 3j - k.
53. 3(a)(i)1 mark· Pure Mathematics · Unit 1 Q3 3(a)(i)Let p = i - j. If q = λi + 2j, find values of λ such that q is parallel to p.
54. 3(a)(i)5 marks· Pure Mathematics · Unit 1 Q3 3(a)(i)Calculate, in degrees, the angle between p and q.
55. 3(a)(i)5 marks· Pure Mathematics · Unit 1 Q3 3(a)(i)Find the value of (a+b)⋅(a-b).
56. 3(a)(ii)2 marks· Pure Mathematics · Unit 1 Q3 3(a)(ii)q is perpendicular to p.
57. 3(a)(ii)5 marks· Pure Mathematics · Unit 1 Q3 3(a)(ii)If 2b - a = 11i, determine the possible values of a and b.
58. 3(a)(ii)a)5 marks· Pure Mathematics · Unit 1 Q3 3(a)(ii)a)Find a non-zero vector v such that p.v = 0.
59. 3(a)(ii)b)1 mark· Pure Mathematics · Unit 1 Q3 3(a)(ii)b)State the relationship between p and v.
60. 3(a)(iii)5 marks· Pure Mathematics · Unit 1 Q3 3(a)(iii)the angle between p and q is π/3.
61. 3(b)(i)a)5 marks· Pure Mathematics · Unit 1 Q3 3(b)(i)a)Calculate, in degrees, the size of the acute angle θ between p and q.
62. 3(b)(i)b)2 marks· Pure Mathematics · Unit 1 Q3 3(b)(i)b)Hence, calculate the area of triangle POQ.
63. 3(b)(ii)a)2 marks· Pure Mathematics · Unit 1 Q3 3(b)(ii)a)Find, in terms of i and j, the position vector of M, where M is the midpoint of PQ.
64. 3(b)(ii)b)3 marks· Pure Mathematics · Unit 1 Q3 3(b)(ii)b)Find, in terms of i and j, the position vector of R, where R is such that PQRO, labelled clockwise, forms a parallelogram.
65. 4(b)9 marks· Pure Mathematics · Unit 1 Q4 4(b)Show that u and v are NOT parallel.
66. 4(b)(i)2 marks· Pure Mathematics · Unit 1 Q4 4(b)(i)Express the vector PQ in the form xi + yj + zk.
67. 4(b)(i)2 marks· Pure Mathematics · Unit 1 Q4 4(b)(i)Express p and q in the form xi + yj.
68. 4(b)(i)3 marks· Pure Mathematics · Unit 1 Q4 4(b)(i)Calculate the lengths of u and v respectively.
69. 4(b)(i)3 marks· Pure Mathematics · Unit 1 Q4 4(b)(i)Determine a unit vector in the direction of OC.
70. 4(b)(i)5 marks· Pure Mathematics · Unit 1 Q4 4(b)(i)Find the values of t such that the vector t r₁ + r₂ is perpendicular to the vector r₂ + r₃.
71. 4(b)(ii)5 marks· Pure Mathematics · Unit 1 Q4 4(b)(ii)Find, by calculation, the position vector of D.
72. 4(b)(ii)4 marks· Pure Mathematics · Unit 1 Q4 4(b)(ii)Find cosθ where θ is the angle between u and v in ℝ³.
73. 4(b)(ii)2 marks· Pure Mathematics · Unit 1 Q4 4(b)(ii)Obtain the vector p - q.
74. 4(b)(ii)6 marks· Pure Mathematics · Unit 1 Q4 4(b)(ii)Determine the Cartesian equation of the plane which passes through the point Q and is perpendicular to PQ.
75. 4(b)(ii)5 marks· Pure Mathematics · Unit 1 Q4 4(b)(ii)By using the identity in (b)(i) above, find the value of θ, 0 ≤ θ ≤ π/2, such that a and b are perpendicular.
76. 4(b)(ii)3 marks· Pure Mathematics · Unit 1 Q4 4(b)(ii)Find the position vector of C relative to O.
77. 4(b)(iii)5 marks· Pure Mathematics · Unit 1 Q4 4(b)(iii)Show that OE and OC are perpendicular.
78. 4(b)(iii)2 marks· Pure Mathematics · Unit 1 Q4 4(b)(iii)Calculate p.q.
79. 4(b)(iii)2 marks· Pure Mathematics · Unit 1 Q4 4(b)(iii)Determine cos AÔC.
80. 4(b)(iv)5 marks· Pure Mathematics · Unit 1 Q4 4(b)(iv)Let the angle between p and q be θ. Use the result of (iii) above to calculate θ in degrees.
81. 4(c)8 marks· Pure Mathematics · Unit 1 Q4 4(c)Find c given that the angle between the vectors a and b is π/3.
82. 4(c)8 marks· Pure Mathematics · Unit 1 Q4 4(c)Determine the angle between OA and OB.
83. 4(c)3 marks· Pure Mathematics · Unit 1 Q4 4(c)Determine the vector equation of a plane which passes through the point (1, 3, 0) and which is perpendicular to the vector 2i + 4j + 5k.
84. 4(c)6 marks· Pure Mathematics · Unit 1 Q4 4(c)Find the position vector of P.
85. 4(c)(i)4 marks· Pure Mathematics · Unit 1 Q4 4(c)(i)Find |AB|.
86. 4(c)(i)3 marks· Pure Mathematics · Unit 1 Q4 4(c)(i)Express the vectors AB and BC in the form xi + yj + zk.
87. 4(c)(i)4 marks· Pure Mathematics · Unit 1 Q4 4(c)(i)Express EACH of the vectors PQ, QR and RP in the form xi + yj + zk.
88. 4(c)(i)5 marks· Pure Mathematics · Unit 1 Q4 4(c)(i)Show that L₁ and L₂ intersect.
89. 4(c)(ii)3 marks· Pure Mathematics · Unit 1 Q4 4(c)(ii)Find the position vector of the mid-point of AB.
90. 4(c)(ii)2 marks· Pure Mathematics · Unit 1 Q4 4(c)(ii)Hence, determine the coordinates of the point of intersection of the two lines.
91. 4(c)(ii)6 marks· Pure Mathematics · Unit 1 Q4 4(c)(ii)Hence, find the value of λ, given that PQR is right-angled with the side PQ as hypotenuse.
92. 4(c)(ii)5 marks· Pure Mathematics · Unit 1 Q4 4(c)(ii)Show that the vector r = − 16j - 8k is perpendicular to the plane through A, B and C.
93. 4(c)(iii)4 marks· Pure Mathematics · Unit 1 Q4 4(c)(iii)Hence, find the Cartesian equation of the plane through A, B and C.
94. 4(d)4 marks· Pure Mathematics · Unit 1 Q4 4(d)Determine the vector equation of a plane which passes through the point (2, 5, 3) and is perpendicular to the vector 4i + 4j - k.