Vectors: A Level Mathematics questions with answers
Free Vectors questions for A Level Mathematics (9709), part of Pure Mathematics 3. Each question has the answer and the marking points an examiner would look for.
The line l has equation r = (3, 1, 1) + λ(−3, −1, 1). The point Q has position vector (−2, 0, 7).
(a)
Find the position vector of the foot of the perpendicular from Q to l.
[5]
(b)
Hence find the perpendicular distance from Q to l.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
(−3, −1, 3)
general point of l B1 vector from Q to general point M1 scalar product with direction = 0 M1 λ = 2 A1 (−3, −1, 3) A1
5
(b)
√18 = 4.24
modulus of the perpendicular vector M1 4.24 A1
2
Question 2Structured · 12 marksA* challenge
The lines l and m have equations l: r = (0, −4, 1) + λ(1, 2, 2) and m: r = (−5, −4, 0) + μ(6, 2, 3). (Vectors are written as (x, y, z) components.)
(a)
Show that l and m intersect and find the position vector of the point of intersection P.
[4]
(b)
Find the acute angle between l and m.
[2]
(c)
The line l′ is the reflection of l in the line m. Find a vector equation of l′.
[6]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
(1, −2, 3)
equate two pairs of components M1 λ = 1, μ = 1 A1 third component verified A1 P = (1, −2, 3) A1
4
(b)
40.4°
cos θ = 163 × 7M1 40.4° A1
2
(c)
r = (1, −2, 3) + s(143, −34, −2)
l′ passes through P B1 takes a point of l, e.g. A(1, −2, 3) + (1, 2, 2) = (2, 0, 5), or unit vectors M1 foot of perpendicular from A to m: P + 1649(6, 2, 3) A1 image A′ = 2F − A (or direction 2(d̂₁·d̂₂)d̂₂ − d̂₁) M1 direction 1147(143, −34, −2) A1 r = (1, −2, 3) + s(143, −34, −2) A1
6
Question 3Structured · 4 marks
Find the angle between the vectors −i − 4j − 3k and 4i − 4j − 4k.
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