A-LevelFurther MathematicsVectors9231/12 · May/June 2016 · Q811 Marks

Find a cartesian equation of the plane Π₁ passing through the points with coordinates (2, −1, 3), (4, 2, -5) and (−1, 3, −2). The plane Π₂ has cartesian equation 3x – y + 2z = 5. Find the acute angle between Π₁ and Π₂. Find a vector equation of the line of intersection of the planes Π₁ and Π₂.

Find a cartesian equation of the plane Π₁ passing through the points with coordinates (2, −1, 3), (4, 2, -5) and (−1, 3, −2).[4]
The plane Π₂ has cartesian equation 3x – y + 2z = 5. Find the acute angle between Π₁ and Π₂.[3]
Find a vector equation of the line of intersection of the planes Π₁ and Π₂.[4]

Mark scheme

Marking points from the official mark scheme (May/June 2016). The mark or mark type is shown in brackets.

[M1A1] AB x AC = (17, 34, 17)
[M1A1] So x + 2y + z = 3
[M1M1] cos θ = (1,2,1)·(3,-1,2) / (√(1²+2²+1²)√(3²+(-1)²+2²)) = 3 / (√6√14) = 3/√84
[A1] ⇒θ = 70.9° or 1.24 radians
[M1A1] Direction of line of intersection is (1,2,1) x (3,-1,2) = (5, 1, -7)
[M1] Finds point common to both planes is (-1,0,4) or (13/7, 4/7, 0) or (0, -1/5, 13/5)
[A1] Equation of line of intersection is r = (-1,0,4) + t(5,1,-7)

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