Answer only one of the following two alternatives. EITHER The lines l₁ and l₂ have equations r = 6i – 3j + s(3i – 4j – 2k) and r = 2i – j – 4k + t(i – 3j – k) respectively. The point P on l₁ and the point Q on l₂ are such that PQ is perpendicular to both l₁ and l₂. Show that the position vector of P is 3i + j + 2k and find the position vector of Q. Find, in the form r = a + λb + µc, an equation of the plane Π which passes through P and is perpendicular to l₁. The plane Π meets the plane r = pi + qj in the line l₃. Find a vector equation of l₃.

Mark scheme

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

1. [M1] Let P be (6+3s, -3-4s, -2s) and Q be (2+t, -1-3t, -4-t). PQ = (2+t-6-3s)i + (-1-3t+3+4s)j + (-4-t+2s)k = (t-3s-4)i + (-3t+4s+2)j + (-t+2s-4)k.
2. [M1A1] PQ is perpendicular to l₁: PQ · (3i - 4j - 2k) = 0. 3(t-3s-4) - 4(-3t+4s+2) - 2(-t+2s-4) = 0 3t-9s-12 + 12t-16s-8 + 2t-4s+8 = 0 17t - 29s - 12 = 0.
3. [M1A1] PQ is perpendicular to l₂: PQ · (i - 3j - k) = 0. (t-3s-4) - 3(-3t+4s+2) - 1(-t+2s-4) = 0 t-3s-4 + 9t-12s-6 + t-2s+4 = 0 11t - 17s - 6 = 0.
4. [M1A1] Solving the simultaneous equations: 17t - 29s = 12 (×17) 11t - 17s = 6 (×29) 289t - 493s = 204 319t - 493s = 174 Subtracting: 30t = -30 => t = -1. Substitute t=-1 into 11t - 17s = 6 => 11(-1) - 17s = 6 => -11 - 17s = 6 => -17s = 17 => s = -1.
5. [A1] Position vector of P = 6i - 3j + (-1)(3i - 4j - 2k) = (6-3)i + (-3+4)j + (0+2)k = 3i + j + 2k (shown). Position vector of Q = 2i - j - 4k + (-1)(i - 3j - k) = (2-1)i + (-1+3)j + (-4+1)k = i + 2j - 3k.
6. [B1] Plane Π is perpendicular to l₁, so its normal vector is the direction vector of l₁, n = (3, -4, -2).
7. [M1A1] Cartesian equation of Π is 3x - 4y - 2z = d. Since P(3, 1, 2) is on Π, 3(3) - 4(1) - 2(2) = d => 9 - 4 - 4 = d => d = 1. Equation of Π: 3x - 4y - 2z = 1.
8. [M1] The plane r = pi + qj is the plane z=0. For the line of intersection, set z=0 in the equation of Π. 3x - 4y - 2(0) = 1 => 3x - 4y = 1.
9. [M1] Parameterize the line: Let y = λ. Then 3x = 1 + 4λ => x = (1/3) + (4/3)λ.
10. [A1A1] Vector equation of l₃: r = (1/3)i + λ(4/3 i + j) or r = (1/3)i + λ(4i+3j) (OE) (e.g. r = (1/3)i + 0j + 0k + λ( (4/3)i + j + 0k ) ).