A-LevelFurther MathematicsVectors9231/12 · Oct/Nov 2011 · Q25 Marks
The position vectors of points A, B, C, relative to the origin O, are a, b, c, where a = 3i + 2j – k, b = 4i – 3j + 2k, c = 3i – j – k.
- (a)Find a × b and deduce the area of the triangle OAB.[3]
- (b)Hence find the volume of the tetrahedron OABC, given that the volume of a tetrahedron is ½ × area of base × perpendicular height.[2]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2011). The mark or mark type is shown in brackets.
- [M1] Finds a × b i j k 3 2 -1 = i - 10j - 17k 4 -3 2
- [A1] Finds a × b i j k 3 2 -1 = i - 10j - 17k 4 -3 2
- [A1] Finds area of base. 1/2 √1² + (−10)² + (−17)² = 1/2 √390 ( = 9.87)
- [M1] Attempts to find height Height = (3i – j – k).(i -10 j −17k) / √1² +(-10)² + (−17)² = 30 / √390 ( = 1.519)
- [A1] Finds volume 1/3 x 1/2 √390 x 30 / √390 = 5
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