Vectors in three dimensions — Question 8
Original QCE Vault practice · 5 marks
Q8 · Practice questionComplex familiar5 marks
QUESTION 8 (5 marks)
A line, l, and a plane, π, are given by
l: r=102+λ12−1,π: 2x−y+z=5.a)Determine the point at which the line intersects the plane.
b)Determine the acute angle between the line and the plane, giving your answer in exact form.
WORKED SOLUTION
Practice marking scheme
5 marksANSWER(0,−2,3);α=sin−1(1/6) Sample response and mark allocation
On the line,
x=1+λ, y=2λ, z=2−λ. Thus
2(1+λ)−2λ+(2−λ)=5.
Substitutes the line into the plane.[1 mark]
λ=−1, giving
(0,−2,3).
Determines the intersection point.[1 mark]
d=(1,2,−1) and
n=(2,−1,1).
Identifies direction and normal vectors.[1 mark]
sinα=∣d∣∣n∣∣d⋅n∣=66∣2−2−1∣=61.
Uses the line–plane angle relationship.[1 mark]
α=sin−1(1/6).
Determines the acute angle.[1 mark]
Practice question aligned to the current QCAA syllabus; review the worked solution and marking criteria.
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