A-LevelFurther MathematicsVectors9231/12 · May/June 2013 · Q1214 Marks
Answer only one of the following two alternatives. OR. The points A, B, C and D have coordinates as follows: A (2, 1, -2), B (4, 1, -1), C (3, -2, -1) and D (3, 6, 2).
- (a)The plane Π₁ passes through the points A, B and C. Find a cartesian equation of Π₁.[4]
- (b)Find the area of triangle ABC and hence, or otherwise, find the volume of the tetrahedron ABCD. [The volume of a tetrahedron is ⅓ × area of base × perpendicular height.][6]
- (c)The plane Π₂ passes through the points A, B and D. Find the acute angle between Π₁ and Π₂.[4]
Mark scheme
Marking points from the official mark scheme (May/June 2013). The mark or mark type is shown in brackets.
- M1A1 Vector perpendicular to Π₁. n₁ = AB x AC = (3, -1, -6)
- M1A1 Obtains cartesian equation. 3x - y - 6z = 17
- M1A1 Obtains area of triangle ABC. 1/2 |AB x AC| = 1/2 √ (3² + (-1)² + (-6)²) = 1/2 √46 (=3.39)
- M1A1 Obtains length of perpendicular from D to triangle ABC. h = |(3(3) - 1(6) - 6(2) - 17)| / √(3² + (-1)² + (-6)²) = |-26| / √46 = 26/√46
- M1A1 Uses 1/3 x Base area x Height. 1/3 * (1/2 √46) * (26/√46) = 13/3
- (Or triple scalar product method.)
- M1A1 Obtains normal to ABD. n₂ = AB x AD = (-5, -7, 10)
- M1A1 Uses scalar product cos θ = |n₁ ⋅ n₂| / (|n₁||n₂|) = |-68| / (√46 * √174) ⇒ θ = 40.5°
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