IB · Mathematics: Analysis & Approaches SL & HL · Geometry & trigonometry

Vectors, lines & planes: IB Mathematics: Analysis & Approaches questions with answers

Free Vectors, lines & planes questions for IB Mathematics: Analysis & Approaches (SL & HL), HL only, part of Geometry & trigonometry. Each question has the answer and the marking points an examiner would look for.

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Vectors, lines & planes practice questions

Question 1Structured · 7 marks

The plane Π has equation 2x − 2z = −12. The line L has equation r = −4−46 + λ−1−21.

(a)

Find the coordinates of the point where L meets Π.

[4]
(b)

Find the shortest distance from the point (0, 0, 0) to the plane Π.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)(−2, 0, 4)
substituting the parametric form of L into the equation of Π M1
−20 − 4λ = −12 A1
λ = −2 A1
(−2, 0, 4) A1
4
(b)12√8
distance = |d| ÷ |n| for the plane r·n = d (M1)
|n| = √8 (A1)
12√8 A1
3
Question 2Structured · 7 marks

The points A, B and C have coordinates A(2, −2, 1), B(2, 0, 3) and C(4, 0, 0).

(a)

Find AB × AC.

[3]
(b)

Hence find the Cartesian equation of the plane containing A, B and C.

[2]
(c)

Find the exact area of triangle ABC.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)−64−4
AB = 022, AC = 22−1 A1
attempt at the cross product (M1)
−64−4 A1
3
(b)−3x + 2y − 2z = −12
using the normal and a point, r·n = a·n (M1)
−3x + 2y − 2z = −12 A1
2
(c)2√172
12|AB × AC| (M1)
12√68 = 2√172 A1
2
Question 3Structured · 9 marks

The line L1 has equation r = −344 + λ110 and the line L2 has equation r = −70−2 + μ112.

(a)

Show that L1 and L2 intersect, and find the coordinates of the point of intersection.

[6]
(b)

Find the cosine of the acute angle between L1 and L2.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)(−4, 3, 4)
equating components M1
two correct equations A1
attempt to solve two equations M1
λ = −1, μ = 3 A1
checking the third equation is satisfied R1
(−4, 3, 4) A1
6
(b)22√3
using the direction vectors (M1)
cos θ = 22√3 (A1)
22√3 A1
3
Question 4Structured · 5 marks

The planes Π1 and Π2 have equations −y + 3z = 4 and 3x + 2y + z = 5.

(a)

Find the acute angle between the planes.

[3]
(b)

Find a vector in the direction of the line of intersection of the planes.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)85.2°
angle between the normals (M1)
cos θ = 12√35 (A1)
85.2° A1
3
(b)−793
n1 × n2 (M1)
−793 A1
2
Question 5Structured · 7 marks

The plane Π has equation −2x − y − 3z = 3. The line L has equation r = −1−1−8 + λ1−1−3.

(a)

Find the coordinates of the point where L meets Π.

[4]
(b)

Find the shortest distance from the point (0, 0, 0) to the plane Π.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)(−4, 2, 1)
substituting the parametric form of L into the equation of Π M1
27 + 8λ = 3 A1
λ = −3 A1
(−4, 2, 1) A1
4
(b)3√14
distance = |d| ÷ |n| for the plane r·n = d (M1)
|n| = √14 (A1)
3√14 A1
3
Question 6Structured · 7 marks

The points A, B and C have coordinates A(1, 3, −2), B(3, 5, 1) and C(0, 2, −3).

(a)

Find AB × AC.

[3]
(b)

Hence find the Cartesian equation of the plane containing A, B and C.

[2]
(c)

Find the exact area of triangle ABC.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)1−10
AB = 223, AC = −1−1−1 A1
attempt at the cross product (M1)
1−10 A1
3
(b)x − y = −2
using the normal and a point, r·n = a·n (M1)
x − y = −2 A1
2
(c)√22
12|AB × AC| (M1)
12√2 = √22 A1
2

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Original IB-style practice questions written for mentorriyaz.in. These are not IB past paper questions, and this site is not affiliated with or endorsed by the International Baccalaureate Organization. Updated 2026-11-04. Paper generator · mentorriyaz.in