A-LevelFurther MathematicsVectors9231/12 · Oct/Nov 2013 · Q810 Marks

The plane Π₁ has equation r = [[2], [3], [-2]] + s[[1], [0], [-1]] + t[[0], [-1], [-2]]. Find a cartesian equation of Π₁. The plane Π₂ has equation 2x – y + z = 10. Find the acute angle between Π₁ and Π₂. Find an equation of the line of intersection of Π₁ and Π₂, giving your answer in the form r = a + λb.

(a)The plane Π₁ has equation r = [[2], [3], [-2]] + s[[1], [0], [-1]] + t[[0], [-1], [-2]]. Find a cartesian equation of Π₁.[3]
(b)The plane Π₂ has equation 2x – y + z = 10. Find the acute angle between Π₁ and Π₂.[2]
(c)Find an equation of the line of intersection of Π₁ and Π₂, giving your answer in the form r = a + λb.[5]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2013). The mark or mark type is shown in brackets.

[M1A1] (a) Finds normal to Π₁. i j k / 1 0 -1 / 0 -1 -2 = i + 3j - k
[A1] Finds Cartesian equation. Equation of Π₁ : x + 3y - z = 12
[M1] (b) Finds angle between normals, using scalar product. cosθ = |2 - 3 - 1| / √(11)√(6)
[A1] = 2/√66 ⇒ θ = 75.7° or 1.32 rad.
[M1A1] (c) Finds direction of line of intersection, using vector product. i j k / 1 3 -1 / 2 -1 1 = 2i - 3j - 7k
[M1A1] Finds point common to both planes. Point on both planes is e.g. (6,2,0)
[A1] States vector equation. r = 6i + 2j + t(2i - 3j - 7k) (OE)

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