A-LevelFurther MathematicsVectors9231/12 · May/June 2020 · Q512 Marks
The lines l₁ and l₂ have equations r = 3i+3k+λ(i+4j+4k) and r = 3i-5j-6k+µ(5j+6k) respectively.
- (a)Find the shortest distance between l₁ and l₂.[5]
- (b)The plane Π contains l₁ and is parallel to the vector i+k. Find the equation of Π, giving your answer in the form ax+by+cz = d.[4]
- (c)Find the acute angle between l₂ and Π.[3]
Mark scheme
Marking points from the official mark scheme (May/June 2020). The mark or mark type is shown in brackets.
- [B1] a₂-a₁ = (0, -5, -9)
- [M1 A1] (1,4,4) × (0,5,6) = (4, -6, 5)
- [M1 A1] (0,-5,-9) ⋅ (4,-6,5) / |(4,-6,5)| = |-9| / √77 = 15/√77 or 1.71
- [M1 A1] Normal vector n = d₁ × (i+k) = (1,4,4) × (1,0,1) = (4,3,-4)
- [M1 A1] r ⋅ (4,3,-4) = (3,0,3) ⋅ (4,3,-4) = 12 - 12 = 0 ⇒ 4x+3y-4z = 0
- [M1 A1FT] sin φ = |d₂ ⋅ n| / (|d₂| |n|) = |(0,5,6) ⋅ (4,3,-4)| / (√(0²+5²+6²) √(4²+3²+(-4)²)) = |-9| / (√61 √41)
- [A1] Acute angle between l₂ and Π is 10.4°
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